Because it is a work of non-fiction, Into the Wild doesn't have a traditional climax; the end of the story is known from the start, and the book is more dedicated to tracing Chris's path and trying to analyze his actions. However, in a purely story-telling sense, the climax comes when Chris decides to return to civilization:
In his journal he now wrote, "Disaster. .. . Rained in. River look impossible. Lonely, scared." He concluded, correctly, that he would probably be swept to his death if he attempted to cross the Teklanika at that place, in those conditions. It would be suicidal; it was simply not an option.
(Krakauer, Into the Wild, Amazon.com)
This incident caused Chris to hike back to the old bus, and the rest of his personal story is that of a man unable to fend for himself anymore. Despite the fact that he had lived more-or-less comfortably for almost four months, he was unable to continue hunting and gathering enough food to live; an experiment in smoking the meat of a moose failed, and so he had no fallback when he began starving to death. His final note, a plea for help, is almost a self-fulfilling prophecy; at that point, it was unlikely that he could have survived on his own no matter what happened.