This is rather the wrong question.
The proper question is: what type of story do you want to tell?
That is, do you want the hero to kill people? Do you need them to kill to advance the story? Do they not like the idea of killing people, similar to batman? If so, forcing them into situations that strain that conviction could work for that. On the other hand, pushing a character that is a killer, that has killed hundreds, thousands, into a situation where they have to reign that in... well, that's when you get things like the new God of War game.
No he isn't. He's a deeply mentally disturbed character, yes... but he knows he is. That's why he has his no killing rule, he knows that he could go over the edge if he crosses that line. It's not unrealistic either, he knows that he'd start justifying things to himself. Does the Joker deserve it? Yes. Does Ras? Probably... What about Mr. Freeze? I don't think so, but... he could and would start justifying that he does deserve death.