#It was the fact Chris and his sister were bastards.

I did partially rewatch the movie and this is explained at approximately 53rd minute.

Christopher's sister narrates that he found out his father had another son from his [father's] first wife to whom he still was officially married **after** Chris was born. Thus, Chris and his sister realized they were bastards.
>This fact suddenly re-defined Chris and me as bastard children. Dad's arrogance made him conveniently oblivious to the pain he caused. And Mom, in the shame and
embarrassment of a young mistress, became his accomplice in deceit.
>
<sub>Source: [Into the Wild script][1]</sub>


  [1]: http://www.script-o-rama.com/movie_scripts/a3/into-the-wild-script-transcript.html