If he enjoys killing that's certainly an issue but not law-breaking. Hunting down criminals with the express purpose of killing them would be however. It's not just a question of innocence but due process. If he's in a fight and they die than he's most likely fine. If he chases down somebody and then just kills them that would be a problem. Plus if that happens too much we'd run out of Rogues :P 

He shouldn't be going Punisher on people but he can justifiably kill someone in combat is what I'm trying to say. At this stage all the characters would be associated with the League but eventually people would be able to branch out.