After seeing it three times--go ahead, be jealous--I really think the scars were somehow given to him by dogs.
There's just so much dog imagery in the movie, so many lines about dogs, allusions to dogs. Even in the end, he's using three dogs.
Just my theory, but I think dogs scarred him.
I think the dog imagery has more to do with the joker being like a dog himself. He even says his MO is like a dog chasing a car.
ps I thought that scene at the end with him standing with the 3 rotties was friggen sweet.