Chris McCandless faced an identity crisis in which he failed to balance realistic and romantic views of himself.
Ultimately, he convinced himself that his book-hero persona Alexander Supertramp could survive the Alaskan outback with a small caliber rifle and a ten pound bag of rice. This act of hubris was a result of mistaking the realism of Jack London and Leo Tolstoy as Romantic cult pulp. Rather than confronting his father's past and his own role in the materialistic American culture at large, McCandless isolated himself in nature to spite mankind. Ironically, he failed to realize how profoundly he impacted even the unlikeliest of strangers (in Ron Franz). His epiphany of "happiness is only real when shared" was realized in the throes of starvation.
Chris McCandless faces several conflicts in the story. The first, and arguably largest is person vs. self. Chris is on a journey of self-discovery throughout the novel. He is constantly searching for what will bring him happiness and fulfillment in life, ultimately giving his life in the process.
But it can also be argued that Chris faces the challenge of person vs. society. It is clear from the back story Krakauer provides that Chris has had trouble fitting into his world since he was young. His sister, Carine remembers him spending much time alone as a child. He doesn't act in a way that would be considered average, in social terms. From spending his Friday nights feeding the homeless, to taking long solo journey's across country Chris struggled to find his place in society, and learn what it meant to be a part of a family for better or worse.
Lastly, and most obviously, Chris faced the conflict of man vs. nature. In the most literal sense Chris struggled to survive in Alaska (as well as the ocean and desert) by foraging, finding shelter, and hunting game. Ultimately this is the conflict that beat him.