There's a lot of different reasons all over the internet and even some insight from his real-life sister, but I recall Christopher mentioning somewhere in the middle of Into the Wild (2007):
- It was hinted all over the first part, but never revealed fully until the middle (approximately).
- It was some "secret" concerning his family and/or his sister.
- He didn't know it at first, but the learned this secret and it was partly responsible for his departure.
- It's not likely to be abuse from his father, as his real-life sister states (though I'm not so sure here).
All in all, I remember that clearly mentioned in the film, but I can't just find that reason.