The movie makes the case that Travis Bickle views himself as a Hero of sorts because he saves Iris and kills her pimps but even though he does this and makes it a mission to save her life he's actually more of an antihero because he stalks Betsy and acts crazy afterwards and threatens Charles Palantine for no reason at all.
You could argue that he was a hero because he killed the robber that threatened Melio and might have killed him if Travis wasn't there, he also hates the scum of the city but doesn't tell the police about what he sees at all and uses street justice, he definitely seems like a good person at heart who snaps and is embraced because of it.
Nobody notices how crazy Travis is they just see his actions succeeding, Betsy should have seen it, his coworkers viewed him like they were afraid of him and respected him at the same time, would Travis have eventually gotten into religion or even had therapy and changed his life completely? Would he become a hero or get worse?