#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 became his accomplice in deceit.
Source: Into the Wild script