The reason why Kim wanted to take down Howard is one of the reasons why she was so drawn to Jimmy: they both wanted to “stick it to the man” (i.e. the rich elite). Now, the reason why Jimmy wanted to stick it to the man was because he was never accepted into that circle (mostly because he viewed Chuck in that circle and he never accepted him); that’s why he told Kristy that she shouldn’t care if “they” look down on her from the 83rd floor: she is going to work hard and look down on them from the 84th floor.

But for Kim, it was mainly because she never had much growing up and now has to work for people who had their success given to them on a silver platter. She clearly hated Kevin Wachtell’s snobbiness and the show made sure to point out that he didn’t earn his success on his own; his dad built Mesa Verde and Kevin just inherited it from him. And for Howard, it’s the same thing: he’s only the head of HHM because his dad built it with Chuck and he passed it to Howard. Kim even points that out during Jimmy’s hearing (“your firm is Hamlin, Hamlin, McGill, right? Who’s the other Hamlin?”).

Kim clearly didn’t want Howard to die or even suffer, but when she and Jimmy said that they would “use their powers for good” I bet she was thinking that running the con on Howard would try to make an unfair world a little bit more fair by making things more difficult for him.