I think he had both scars done to him by a 3rd party, maybe they thought it was ironic. He's always making bad jokes, giving people a pained smile. Do we get to find out what didn't kill him, what made him stranger? Why would Nolan mention it if there wasn't at least a reference to it in TDK? Maybe the Mob did him in, that's why he's targeting the Mob. The Wizard article said:
"Reports indicate the Joker begins the film as a nobody thug before an unknown attack results in his disfigurement thanks to a razor blade. Left for dead, the encounter alters the mans perceptions resulting in the birth of Batmans greatest villain.
But I doubt we'll see that much, but it might explain his obsession with knives.