The pure nature of the Joker, especially in the movie sense, is to be completely chaotic wanting "to watch the world burn". In other words, the Joker is the very definition of the alignment: Chaotic Evil. read more
To expand on his identity as the chaotic antagonist a little further, in relation to the title question, it may well be that the Joker gave himself the scars just because he could. There's no reason that his appetite for destruction and chaos should be limited to external entities - as Arthur says, "some men just want to watch the world burn". read more
In the movie TDK, Chirstopher Nolan has maintained the sanctity of Joker's character. Just like in the comics, he has kept The Joker's origin and identity a mystery. Similar to the original story, he presents different reasons for the scar. read more
His dad was furious and took the blade and carved it across his face. He then took the Joker's mom's favorite lipstick and smeared it on the wound, making it incredibly painful. The Joker then bled internally until a woman named Joyce helped him after hearing his cries. read more