Katsuki Bakugo is one of the main characters in Kohei Horikoshi's My Hero Academia. Introduced in the beginning as a bully who often made fun of Deku, the protagonist, Bakugo aims for the same goal as the guy he liked to torment so much, that is, to become the greatest Hero of all time.
Just like Deku, he was enamored by the charm of All Might and wanted to defeat villains just like him in the future. Since then, Bakugo has worked towards his goal with quite a lot of hard work and, currently, he's among the greatest Hero aspirants in the series, surpassing even some Pro Heroes already. Bakugo has achieved quite a lot so far, with more to come in the future.
7 First In UA Entrance Exam
Bakugo is an excellent user of his powers and, as far as becoming a strong Hero goes, he is certainly a fitting candidate. While the likes of Deku struggled in the UA entrance exam, Bakugo aced it and actually managed to secure the first position.
He did this without even breaking a sweat and that certainly goes to show that he is in a league of his own when compared to the other students, at least at the beginning of the story.
6 First In UA Sports Festival
Just as he finished first in the UA entrance exam, Bakugo also secured first place during the Sports Festival. During the sports festival, he showed his combat ability to the entirety of Japan and when pit against his classmates and other Hero aspirants, Bakugo showed that he was, in fact, the greatest.
Not only did he defeat his classmates, such as Tokoyami, but he also ended up fighting those who are known to hold great potential, such as Todoroki. He defeated Todoroki in the finals and secured first place for himself. He has only grown in skill and might since then.
5 Wasn't Swayed By Villains
Bakugo always had villainous tendencies and, seeing that, the League of Villains ended up targeting him. When kidnapped, the League tried everything they possibly could to sway him to their side. However, Bakugo was not swayed. He laughed in their face and made it known to them that he was already swayed by All Might as a child and that he would eventually go on to become a Hero as great as him in the future.
Doing so in the face of death is certainly admirable and an achievement for him, given that anyone else would have succumbed to the offer to save their lives immediately.
4 Saved Deku
As a Hero aspirant, the utmost priority for Bakugo is to defeat villains and all those who stand in his way. He likes crushing his enemies and breaking their spirits and he does all this to become as great as All Might. That certainly does not mean that Bakugo doesn't rescue people. Rescuing those in need with a smile is a big part of becoming a Hero and Bakugo certainly does it when needed, although his priority is usually crushing the enemies.
One of the biggest saves that Bakugo has ever made in My Hero Academia came to the fans as a shock during the Paranormal Liberation War arc. In this war, Bakugo saved Deku's life by jumping in front of an attack launched by All For One that could have killed his friend. Bakugo did not do this consciously but his body moved on its own seeing that his friend was in danger. That is a true sign of being a Hero and certainly a great character development moment and achievement for Bakugo.
3 Apologized To Deku
Bakugo's character has continued to progress throughout the arcs. When Deku went rogue, Bakugo ended up apologizing to him for all the years of bad behavior. As a child, Deku was bullied by Bakugo and even when the two were in UA together, the latter kept making him feel inferior.
This behavior has stopped now, after Bakugo apologized to him from the bottom of his heart. At the end of the day, the two are friends and Bakugo is now trying his utmost best to enhance their relationship, even though he still aims to beat him to the throne of the greatest Hero of all time.
2 Competes With Deku
One of the biggest achievements of Bakugo is certainly keeping up with Deku. While Bakugo was born with the Explosion Quirk, which is great, his powers don't hold up when compared to the One For All, which is the Quirk that was formerly wielded by All Might.
What's more, it only gets stronger when passed down from generation to generation. As such, the One For All that Deku wields is stronger than the one that All Might possessed. Furthermore, Deku's One For All is superior as it also has six further Quirks inside it that he can access in battle. The fact that Bakugo can keep up with that is surely an achievement.
1 Fought Shigaraki Tomura
In the final war of My Hero Academia, Bakugo ended up fighting against Shigaraki Tomura. This was a fight where he took on someone much stronger than himself and, despite that, he did very well. He unleashed his new powers against him and was able to damage him, which is nothing short of incredible.
He also ended up receiving a gruesome injury and was on death's door shortly afterwards, however, the fact that he put up a fight against the monster that is Shigaraki, is certainly an achievement in itself.