Spoiler alert
I can feel the hate coming from the Togami fans but I must push that to the side and focus on why I hate Felicia.
He has no reason to treat the killing like a game
Now note I have not seen anyone play Danganronpa 1 so any Togami free time events have been thrown out the window because I watch the animation.  There are 2 characters in the Danganronpa series who believe the killings to just be a game.  These characters are Celeste and Togami.  Celeste however being the gambler she is has a reason for this because if she continued to think the killings are a game then she is slightly more comfortable in her circumstances.  Togami on the other hand is just doesn't have a reason as he repeatedly messes with crime scenes and tweeting everyone even people he seemed kinda close to like Celeste as if they were chips in a game of poker between him and Monokuma.
Messing with crime scenes
I could feel with a lot of things but when Togami messed with Fujisaki's corpse.  He could have gotten everyone killed and even himself and yes I'm sure he would go and die if the killer wasn't caught as he seemed perfectly content to not tell anyone about messing with the crime scene.
His personality
He's a fighter but he's also arrogant and rude so no I don't like his personality.
The way he treats Fukawa
I'm sure everyone knows that he treats Fukawa like she's absolutely nothing and that just bothers me so much.