Since he discovered his quirk in his childhood, Katsuki has been the star of his social circle both between his small group of friends and in school. That all changed with U.A., between being surrounded by equally gifted students and leaving a bad first impression, Katsuki lost his role as the classroom star. The initial Battle Trial arc which saw him threaten Izuku’s health instead of focusing on the test at hand soured his relationship with the rest of his class. From that point on, he was viewed as being hyper-aggressive by his classmates, isolated, and even teased for being anger prone.

His aggressive behavior reflected badly on his future as a licensed hero. He took an internship with the pro hero Best Jeanist and discovered that he doesn’t have a positive public perception. Furthermore, his performance at the U.A. Sports Festival painted him in such a negative light that the League of Villains kidnapped him because they saw in him the potential to be turned into a villain.

Over time, Katsuki had become a much calmer and reserved student when he’s not provoked. It seems that the many recent incidents he’s been a part of have begun affecting his attitude towards the people around him for the better. This is best reflected when Class 1-A moves into the new U.A. High School dorms and are in a somber mood after criticism from professor Shota Aizawa. Katsuki activates classmate Kaminari’s quirk which makes his brain short-circuit and gets the rest of his class laughing. It’s his first genuine positive interaction with his classmates and a sign of improved relations, but there’s still a gap between him and Class 1-A as he ditched them at the start of the Provisional Hero License Test to go off on his own.

One of the best-developed friendships in My Hero Academia is the one between Katsuki and Eijiro. He was initially very critical of Katsuki’s hyper-aggressive behavior and asked All Might to put an end to Katsuki’s Battle Trial because of what he might do to Izuku. Despite this rocky start, Eijiro and Katsuki’s friendship has grown progressively since. Their friendship truly begins in the Unforeseen Simulation arc when they’re both teleported to the same simulation zone and work together to easily dispatch the villains sent to take them out.

They renew their cooperation at the Sports Festival when Eijiro overlooks Katsuki forgetting his name and convinces him that he would be the perfect mount, as well as matches his ambition to take out the number one target in the cavalry battle, Izuku. Their success together and Eijiro’s matching determination is enough for Katsuki to give Eijiro the respect he deserves by calling him by his name.

Later, he helps Eijiro pass the Final Exams, and they run off together at the start of the Provisional Hero License test. The crowning moment of their friendship is also a crowning moment for Katsuki’s evolution as a character. Izuku contrives a plan to save Katsuki so that he isn’t a distraction in All Might’s fight and realizes that the only person Katsuki could possibly accept getting help from is Eijiro. In an extremely well animated moment, Katsuki blasts across the sky to grab Eijiro’s hand and escape the wrath of the League of Villains.

While Katsuki has certainly become a better person over My Hero Academia’s first 53 episodes, his relationship with Izuku remains rocky. It’s his bullying of Izuku which initially painted him as a bad person, and it hasn’t really softened across the series. In the Battle Trial arc, he attempted to severely injure Izuku, and during the Final Exam arc, he had a hard time cooperating with him to take down All Might. It’s hard to tell where the source of Katsuki’s hatred for Izuku comes from, but a pivotal point in their relationship appears to be a quirkless Izuku trying to help him after he fell from a makeshift bridge. Repeated situations where Katsuki was in a position of weakness and Izuku tried to help him have further sullied any possibility of mending their childhood friendship. We can at least say that Katsuki no longer goes out of his way to clash with Izuku, but it’s clear that he sees him as a rival and someone he cannot allow to surpass him.