Powerful scene. Since the wounds were from behind, they were likely not self-inflicted. It's possible that he was playing with a friend who accidentally shot him.
But there's no mention of a friend - only a father. Since the ghosts in The Sixth Sense usually exist to clear a severe disturbing injustice, it's also possible that his father (drunk and abusive) shot him.
Can't believe that I'm comparing a child to an educated doctor of child psychology, but much like Dr. Crowe, this child doesn't know that he's dead and been dead since the 1970's and obviously his friend is the one who shot him because you can see the entry wound for the bullet [it's the same size as Dr. Crowe's entry wound for a bullet] and it's underneath his bangs with his hair parted just slightly off and then the back of his head is completely blown off. He is the only other ghost besides Dr. Crowe who doesn't realize he's already dead, he used to live in Lynn Sear's house back in the 1970's when he was a teenager and he thinks he still lives there; he thinks his friend is still there spending the night with him and being left home alone by his parents when he gets the stupid idea to play with daddy's gun. The friend shot him because the angle of the entrance wound cannot lend itself to be self inflicted. I wonder what happened to that teenage boy ghost since he was the only one who was calm, docile and friendly towards Cole and didn't know he was already dead.