Katsuki and Izuku were friends during their childhood, however once Katsuki learned Izuku didn't have a Quirk while he himself manifested a powerful one, he had always considered Izuku as a hindrance in his path to becoming the greatest hero, bullying Izuku alongside his classmates and giving him the derogatory name "Deku" which he always calls him. In return, Izuku always refers to Katsuki as "Kacchan".
In their adolescence, Katsuki continued to bully Izuku even burning one of his notebooks when learning he was going to the same high school. However, he was surprised when Izuku ran into danger to save him from the Sludge Villain inspiring All Might to take action in spite of his time limit. Katsuki caught up with Izuku and berated him for his actions, calling him a Quirkless failure most likely out of pride since he acknowledged he owed him. At U.A., Katsuki still looked down on Izuku, but was shocked to see that Midoriya had developed a powerful Quirk. This sight convinced him that Izuku was "hiding this power from him". He nearly attacked him for answers before he was restrained by Shota Aizawa. As he felt superior to Izuku, he overlooked Izuku's intelligence and ingenuity both of which Izuku used along with his quirk to finally stand up to Katsuki in their mock battle. Due to this, Katsuki's ego caused him to believe Izuku was making fun of him even believing in one instance Izuku taunted him such as when Izuku tried to save him after Katsuki fell into a river. Katsuki attacked Izuku and insulted him, as he demanded that he used his quirk in their fight. When Izuku finally used his quirk against him, Katsuki was ready to prove himself only to realize Izuku had another plan and defeated him for the first time.
After his loss to Izuku in the Battle Trial, Katsuki was in disbelief. His defeat, along with witnessing how talented the rest of his class was in comparison to himself left him frustrated. Afterward, Izuku made an out-of-nowhere confession of possessing a borrowed quirk which Katsuki seemingly did not believe. During that encounter, Izuku swore to beat him. He has always shown hostility towards Izuku whenever he interacts with him. Katsuki's antagonism towards Izuku only intensified after seeing Izuku improved on his use of his Quirk as well as using his moves to help. When teamed up with Izuku, Katsuki ignored whatever he suggested and went as far as to attack him for trying to help him and stated that he doesn't need his help. However, after Katsuki stated to All Might he would rather lose than accept help from others, an irritated Izuku (surprisingly) punched him in the face stating he's not the type to just give up like that. Though reluctant, Katsuki got the message and finally worked together with Izuku and willingly protected him from All Might so he could get to the finish line.
However, as pointed out by All Might and Ochaco, Katsuki is actually jealous of Izuku's growth and the potential to become stronger than him. He is also jealous of Izuku's popularity with the rest of the class. Katsuki, however furiously denies such statements. His second fight with Izuku, however would prove that these statements are indeed true, and that Katsuki would begrudgingly acknowledge he's been jealous of Izuku's quick growth.
After Katsuki was kidnapped by the League of Villains, Izuku was part of the Rescue Bakugo Team.
After the Hero License Exam, Katsuki called out Izuku to meet him at Ground Beta where they first fought. There, Katsuki told Izuku his deduction on All Might and All For One and correctly deduced Izuku's Quirk was given to him by All Might. After Izuku confirmed it, Katsuki attacked Izuku, proclaiming they were going to fight since he wanted to see his potential. Despite growing up together and idolizing the same person, Katsuki did not understand why All Might chose Izuku over himself. Izuku realized while they fought that even if the fight seemed meaningless on it's own, Katsuki needed it. During their encounter, Katsuki confides in Izuku that he blamed his needing to be rescued as the reason for All Might's early retirement.
After the fight, All Might explained to Katsuki about his power and Katsuki became one of the few to know the truth of Izuku's quirk. With this, it seems that, no matter how much he hates it, Katsuki does at least acknowledge Izuku's abilities. Katsuki seems to at least respect Izuku as a fighter and a rival, going as far as to point out a weakness of the latter's Shoot Style.
Interestingly, despite his malice towards Izuku, Katsuki has on two occasions expressed his true feelings in front of him. Once, after admitting defeat to Izuku and his realization of how 'powerful' he truly was at U.A. and the second occasion when he confessed his anguish towards All Might's current state. Katsuki learns to see just how fearful he is of Izuku's rapid development and how much they could accomplish if they were to work together, and from there, the two slowly start to build their friendship back up.
Other than Izuku, Katsuki interacts with Eijiro another one of his classmates. Katsuki seems to get along well with Eijiro, despite their differing personalities.
Eijiro initially didn't like Katsuki's explosive attitude though he began to change his mind in the Unforeseen Simulation Joint Arc when they teamed up. Since then Eijiro has grown to respect Katsuki, interpreting his actions as manly, something Eijiro appears to highly value. Although, he's not above getting frustrated with Katsuki's tendencies, as seen when Eijiro tried to get him to cool off after Monoma insulted them during the cavalry battle.
Although Katsuki at first didn't give Eijiro much attention, he seems to appreciate their Quirks' compatibility with each other, as he accepted Eijiro's proposal to form a team during the Human Cavalry Battle after smiling at his plan to take down Izuku. However, at the beginning of the Sports Festival, Katsuki called Eijiro "hedgehog hair" (despite having spiky hair himself which got pointed out to him), which irritated Eijiro. Katsuki later called Eijiro by his last name, showing he has come to respect or acknowledge him.
Their relationship better improves as the series progresses, and, overall, Katsuki gets along with Eijiro. As such, Eijiro is the only person in Class 1-A Katsuki actually respects and would possibly go as far as being called a friend. This can be seen as Eijiro went to Katsuki for studying for the mid-terms since the latter scored higher than him. As noted by Izuku, Eijiro is the only one in their class who can call out to help Katsuki without hurting Katsuki's pride, which was proven true as Katsuki grabbed his hand when he was rescued.
During the Hero License Exam, Denki noted that Katsuki had reduced the power of his explosions to avoid harming Eijiro and himself. Eijiro asked Katsuki for help in studying, which they did together for the end of the term exams.
When Eijiro was sad about almost getting he and his classmates expelled, Katsuki tried cheering him up by paying him back for the 50,000 yen night vision goggles Eijiro bought to rescue Katsuki. Eijiro openly voices his self-doubts, Katsuki reminds him that he had proclaimed himself as an unwavering horse during the Sports Festival and said if Eijiro refuses to go down, that proves how strong he is.
Katsuki insults Kaminari frequently, who returns the sentiment by openly teasing Katsuki shown on the bus ride to USJ which angered the latter.
Despite his teasing, Kaminari was intimidated by Katsuki in Final Exams Arc but agreed with the latter's point of mastering his quirk.
Later on, as some even got away with calling him Kacchan, a name Izuku calls the latter. It seems they have a friendly relationship as they both get along with Kirishima and seem to work together well.
When the students were moving into dorms and Aizawa told the class about how if it hadn't been for All Might's retirement, he would have expelled everyone except Bakugo, Jirou and Hagakure, Bakugo forcibly activates Kaminari's quirk to cause him to go into his brainless state and lighten the mood.
Having witnessed his power, Katsuki was greatly shocked at Shoto's potential and began to doubt his capabilities.
Katsuki sees Shoto as another rival, and someone that he needs to defeat to become Number One. On the other hand, Shoto has occasionally has been shown not to be scared of Katsuki in the slightest, casually making comments that could tick the other off.
During the U.A. Sports Festival Arc, when Shoto and Izuku talked to each other after the Human Cavalry Battle, Katsuki secretly overheard Shoto's backstory, with a seemingly surprised expression on his face. Despite hearing Shoto's tragic backstory, Katsuki showed no sympathy towards him, although his statement about not caring about Shoto's problems or feelings could also be taken to mean he wanted the two of them to be able to fight without being held back by anything. After their battle in the Sports Festival, Katsuki despised Shoto for not giving him a battle worthy enough of being called Number One. Katsuki's anger caused him to be fully restrained during the closing ceremony of the Sports Festival so he wouldn't release his fury onto Shoto. In the English dub, his nickname for Shoto is "Icy Hot".
During the Final Exams Arc, he made a promise to defeat Shoto after he just swore to take Izuku down as well. In this statement, he told the latter he won't have another half-effort win like at their previous match.
However, Katsuki seems to have let go of his grudge or put aside for a while as the two were seen having a normal conversation during their training at the lodge cabin though he didn't want to be paired up with him and wanted Ojiro to trade with him. In other words, Katsuki and Shoto appear to be somewhat neutral with each other, with the latter not hesitating to interact with Katsuki in a relaxed way.
Shoto also joined some of his classmates in saving Katsuki from the League of Villains. It is unknown what sort of standing they have now since this experience. Though during the Remedial Course Arc, while trying to tame the bad behaving kids, Katsuki mentions they need some violence as that is how his parents raised him. Shoto ensured him there was a better way. Katsuki thought back to Shoto's past and then let the latter make the plan, showing he is at least sensitive to Shoto's past.
Katsuki doesn't bother to interact with Ochaco because of her friendship with Izuku. Though they first saw each other in their mock battle, Katsuki was so focused on fighting Izuku that he ignored Ochaco.
They fully interact during the Sports Festival tournament, when they have to face each other in the first round. Ochaco was utterly terrified at the prospect of having to fight Katsuki, while he thought very little of her. At the beginning of their battle he nicknamed her "round face." Throughout their battle, Katsuki in his own way, had offered Ochaco the chance to withdraw since he would show her no mercy even if she is a girl. When Ochaco prepared a secret attack on him, Katsuki was surprised by her ingenuity though overcame it without a problem before he recognized her as one of Izuku's friends and expected this from her.
At the end of their match, Katsuki used her last name before finishing her off, indicating that Katsuki respects Ochaco's perseverance (as he usually doesn't use peoples' names save for those who impress him or has a rivarly). Katsuki went out of his way to say that Ochaco wasn't fragile, when defending himself for fighting her at full strength.
In an omake, Ochaco talked with Katsuki about his relationship with Izuku. She questions why he is so angry around him and correctly guesses his anger is really just a way to mask his insecurities and jealousy towards Izuku. She suggests they should start being friends again, given their past. Katsuki responds with anger.
Because of their equally forceful personalities and very different personal values, Katsuki and Tenya don't get along well. Katsuki looks down on Tenya's privileged background, and dislikes being ordered around by him, whereas Tenya disapproves and frequently reprimands Katsuki for his improper behavior, questioning his intentions to become a hero. This was especially apparent when they were forced to partner up for the Trial of Battle.
Despite their bad standing, Tenya joined some of his classmates in saving Katsuki from the League of Villains. It is unknown what sort of standing they have now since this experience. For the most part, they don't seem to be on bad terms even though Katsuki got irritated with him reprimanding him for fighting Izuku.
Similarly to Izuku and Shoto, Katsuki idolized All Might as a child. While very little of this respect and admiration is apparent in present time, it seemed to have a strong impact on Katsuki's belief in what a hero should be.
He first met the famed hero in person when the latter saved Katsuki after he was attacked by the Sludge Villain.
They met again at the U.A. High School, wherein the training match between Izuku and Katsuki, All Might scolded Katsuki his reckless and cruel fighting style that could have killed Izuku.
Despite Katsuki's aggressive nature, All Might acknowledges him as a promising hero but has repeatedly attempted to guide him whenever possible. Although his advice has often fallen on deaf ears and All Might is often unsettled by Katsuki's behavior, be it his aggression or his reckless drive.
Unfortunately, despite his past admiration for the renowned hero, Katsuki does not hesitate to either insult or challenge All Might to a fight, refusing to run away when he was assigned to fight him, only to suffer a humiliating defeat, just as Izuku stated. He was also angry and embarrassed that he had to be saved by All Might when he was kidnapped and even refused to thank him. Despite his antagonistic attitude towards his former idol, Katsuki did show concern when All For One attacked him, showing that he at least somewhat care for him.
After seeing his true form, he still believed in All Might's ability to be a hero and refused to lose his faith in him. However, with All Might in retirement, Katsuki seems to have gotten a clue about All Might's relationship with Izuku, going as far as to ask him about what he is to him. Even still, Katsuki did thank All Might for saving him from the League of Villains. Which further states his intellect.
In the end, Katsuki's admiration and concern for All Might were revealed to be larger than he had previously shown. During his fight with Izuku, he admitted that he blamed himself for All Might's retirement and current state after being kidnapped and after the battle with All For One. This had left a lasting impact on him, as he did not know what to do after the fact.
Having watched the whole sight, All Might comforted Katsuki after his battle with Izuku ended, he told Katsuki that he is not responsible for his pitiful state and apologized to him as well as told him the truth about One For All. Katsuki understood and promised to keep the latter's secret from others becoming one of the few to the truth of All Might's powers.
Shota is Katsuki's homeroom teacher and they barely get along, due to Katsuki's rude and violent behavior. Shota once restrained Katsuki for nearly attacking Izuku and tends to look over Katsuki more than any other student due to his violent and irresponsible behavior.
However, Katsuki appears to respect his teacher for being a hero while Shota acknowledges he has the qualities of a hero when asked by the press about his "villain like" behavior.
When he met Bakugo Family, he was complimented for seeing Katsuki for he was and not overlooking his faults.
Katsuki selects Tsunagu for his internship. When Tsunagu styles his hair, he realizes he made a bad choice working for him. Though Tsunagu tried to teach Katsuki the qualities of a hero by learning to quell his anger, Katsuki didn't appear to listen to any of his lectures. Katsuki even considered his time with Tsunagu to be worthless as he didn't improve in skill like Izuku did.
Hanta and Katsuki tend to be in close proximity due to their friendships with Eijiro. Hanta even mentioned that nobody would actually vote for Katsuki as representative which annoyed Katsuki.
They were in the same team during the cavalry battle, where it was demonstrated that Hanta often gets exasperated with Katsuki's extreme competitiveness, but is willing to cooperate with him regardless. Katsuki refers to him as "Tape Arms" or "Soy Sauce Face".
Katsuki didn't know of him so he didn't give Neito much attention until he took his bandana and mocked him for being attacked by the Sludge Villain and his previous speech of winning the festival. This mockery alone was enough to enrage Katsuki so much to the point of overlooking Izuku, whom he had his eyes on, in order to destroy Neito's team first. Later, after repeated taunting, mocking and copying of his Quirk, Katsuki's rage just grew even further and he swore again that he will become number 1, but this time he adds that he will be destroying everyone.
In the end, Katsuki accomplished his goal and beat Neito's team and leaving the latter scared of him.
Katsuki first met Tomura in the Unforeseen Simulation Joint Arc. He was nearly attacked by the villain before All Might appeared. During the Final Exams Arc, Katsuki learned the League of Villains made plans to procure him, which was successful.
Tomura then offered him a place in his organization since they both want to do what they want, but Katsuki refused as he voiced his admiration to All Might and was intent on being a hero. Katsuki even attacked Tomura and stood defiant at the latter's possible retaliation. At the end, Katsuki was rescued by his classmates and likely considers Tomura an enemy.
Katsuki doesn't seem to hold much respect for his mother, as he insulted and threatened her in front of his teachers, however these threats are more like his usual, childish reactions to everything and do not seem genuine.
His mother has a similar attitude towards her son, as she often yells back at him and insults him too. She is aware of her son's terrible behavior and disapproves of it. She blames this on the fact that her son has a strong Quirk, and has always been talented to everything he has ever attended and received a lot of praise for it, which caused him to develop his bad attitude. These words seem to make Katsuki angry at her since they are most belittling him. Katsuki's mother approves of her son going to U.A though, as she believes the school will see him for who he truly is. Mitsuki is one of the only people who can really talk back to Bakugo without him exploding in rage.
Katsuki also shows some kind of fear towards his mother after she hits him, as he trembles when fighting with her.
All Might also comment that their family is the most dysfunctional one he has ever seen.
Similar to his mother, Mitsuki Bakugo, Katsuki does not appear to hold much respect for him even telling his own father to be quiet in front of his teachers. However, these threats are more like his usual, childish reactions to everything and not genuine.
Unlike Katsuki's mother, his father has a rather calm personality, and tries to get through to his son, but his short temper prevents it. Like his wife, he clearly loves his son very much and does not approve of his son's arrogant attitude.
Katsuki could see through Yo's nice façade and finds him fake and aggravating, causing Yo to smirk. The two later came to blows in the Provisional Hero License Exam.