To understand Chris's journeying, we need to distinguish between a tourist and a pilgrim. A tourist travels for novelty, to see new sights, and to have a break from everyday life. Often the goal is simply pleasure. The tourist may come home with a thousand photos and a pile of souvenirs, but he doesn't expect to be profoundly changed by his journey.

The pilgrim, on the other hand, travels for the purpose of spiritual transformation. He wants not to consume his new surroundings but to be changed by them. He journeys not for superficial pleasure but to more fully encounter both the divine and his own soul.

Much of what Chris wrote and did reveal a pilgrim, a person seeking soul transformation. Chris did not desire superficial pleasure. He sought the deep joy, often accompanied by hunger or physical discomfort (the opposite of pleasure), that comes when we live life fully. As he writes to Ron:

The joy of life comes from our encounters with new experiences, and hence there is no greater joy than to have an endlessly changing horizon, for each day to have a new and different sun.

More to the point, Chris explains that he is wandering not just for the sake of wandering but to be transformed, to "move into an entirely new realm of experience." He says:

Don't settle down and sit in one place. Move around, be nomadic, make each day a new horizon. You are still going to live a long time, Ron, and it would be a shame if you did not take the opportunity to revolutionize your life and move into an entirely new realm of experience.

Finally, Krakauer describes Chris's final destination, in which he braved the Alaskan wilderness all alone, as follows:

The trip was to be an odyssey in the fullest sense of the word, an epic journey that would change everything.

In other words, Chris travels to Alaska as a pilgrim, seeking transformation.

Many quotes from the book show Chris to have been a pilgrim, a person on a spiritual quest trying to come closer to what is most essential in life.

I can't prove that he is a pilgrim, but I can give some supporting quotes to that concept.  About halfway through the book, author John Krakauer contemplates the mystery that is Chris McCandless.  He writes the following quote:

McCandless was something else—although precisely what is hard to say. A pilgrim, perhaps.

Krakauer then weaves in further hints that McCandless is a pilgrim or very much like historical figures that were pilgrims.  Take this one for example:

His ambivalence toward sex echoes that of celebrated others who embraced wilderness with single-minded passion—Thoreau (who was a lifelong virgin) and the naturalist John Muir, most prominently— to say nothing of countless lesser-known pilgrims, seekers, misfits, and adventurers.

The above quote explicitly links McCandless with other wilderness pioneers/pilgrims.  Thoreau and Muir weren't anti-society, but they most definitely found an inner peace by existing in harmony with nature and the land.  They found it spiritual, and that is what McCandless felt as well.  The following quote is from McCandless in his letter to Ronald Franz.  

The very basic core of a man’s living spirit is his passion for adventure. The joy of life comes from our encounters with new experiences, and hence there is no greater joy than to have an endlessly changing horizon, for each day to have a new and different sun.

Another quote that shows Krakauer directly linking McCandless with pilgrims is the following quote from Paul Shepherd.  

To the desert go prophets and hermits; through deserts go pilgrims and exiles.

At the time the quote appears in the book, the quote doesn't seem to apply to McCandless; however, at the end of Chapter Four, it is obvious that McCandless is the desert pilgrim that the quote speaks about.