We first meet Hawkeye as he lounges by the banks of a stream with his Indian friend Chingachgook, cool as a cucumber. He's clothed in the frontier style: hunting shirt, fur cap, buckskin leggings, and Indian moccasins, and he carries a few weapons, the most striking of which is the "long rifle" that has earned him his (awesome) nickname "la Longue carabine." The narrator writes:
Notwithstanding the symptoms of habitual suspicion, his countenance was not only without guile, but at the moment at which he is introduced, it was charged with an expression of sturdy honesty. (2.3)
Translation: he's a good guy. He's a stand-up dude. He's all that and a bag of chips. He's the mascot of a Big 10 football team for Pete's sake.
And it's not too long into the novel before we also know he's not somebody to mess with. You do not cross Hawkeye. Dude is loyal to his friends, a superior marksman, and a fantastic guide in the wilderness: he's the coolest. He dresses up as a bear. He has insane rifle skills. He makes the Dos Equis guy look like a chump. The Munro sisters and Heyward lucked out hardcore when they convince him to serve as their escort.
In many respects, Hawkeye acts like a Native American and adopts Native American values. He often scoffs at Heyward's lily-white notions of military etiquette and David Gamut's book learning while frequently emulating his Mohican buds: he knows the wilderness and refers to his experience in the forest as the only teacher he really needs.
At the same time, however, Hawkeye emphasizes over and over again that he is a "man without a cross," meaning that his white blood has never mingled with another race's. If he were in the world of Harry Potter he'd be throwing around the term "mudblood" like there was no tomorrow. His blood is "pure" (blegh) and he often uses this to explain some of his actions that deviate from the stereotypical Native American norm.
For example, Indians are portrayed in the novel as inherently savage with an almost irresistible urge to kill and scalp their enemies (boo, Cooper. Really?). Hawkeye explains his decision to spare the lives of Magua and the Huron conjurer as a product of his pure white blood.
Hawkeye, then, represents a white man who has stripped away useless or inessential aspects of white civilization, such as antiquated notions of honor (exemplified by Heyward) and formal education (exemplified by David Gamut), in order to successfully survive in the wilderness. He retains a supposedly essential "whiteness" in his nature that cannot be altered.
No analysis of Hawkeye would be complete without comparing him to his BFF Chingachgook. The two men take great pride in their "unmixed" blood, and a case can be made for them as Native American/white doppelgangers.
Both are experienced foresters, both are apparently celibate, and both are about the same age. There is an important difference, however, that can be summed up by the title The Last of the Mohicans. Chingachgook is the last of his line. He represents a way of life that is slowly dying out as they are squeezed out of their land by the "pale faces." Chingachgook embodies the last of the Mohicans while Hawkeye embodies the first of the new type of white American man—one who has left useless European traditions behind in order to survive on the frontier.