In Cooper's mind, your racial makeup determines how you act, what you believe, and whom you fall in love with. You can take on the attributes of another race (like Hawkeye), but there is an irreducible element within you that will resist all attempts at cultural conversion—like Hawkeye's "whiteness."
Hawkeye is such an important character—and the protagonist of the story—because he successfully adopts Native American ways in order to survive in the wilderness. With Hawkeye, Cooper shows his audience that it is possible to learn from Indians without compromising whiteness.
Remember that Cooper was writing in the 1820s, a time of territorial expansion. Issues of the frontier—and how to survive on it—were incredibly important, as well as issues like slavery and Native American relations.
Maybe it's because of this that Cooper shows biracial characters as existing in a kind of dangerous in-between land. Cora, who is part African-American, gazes at Magua in the opening scenes of the novel in "pity, admiration, and horror," (1.21) and remarks: "Who that looks at this creature of nature, remembers the shade of his skin?" (6.6) when she's checking out Uncas.
Cora should be the ideal here—she's able to pity the (pretty pitiable) Magua, and she looks past Uncas' race to see what is really there: a super-hawt dude. But Cooper portrays her as having had her whiteness diluted and therefore existing in a twilight state. She's killed by the end of the novel, and it kind of seems like this is her punishment for looking past race and skin color.