According to Hurston, through what methods should African Americans attain their freedom and recognition? What role does black art, cultural expression, and creativity play?

"How It Feels to Be Colored Me," published in 1928, is a personal essay based both on Hurston's upbringing in Eatonville, Florida and part of her adulthood in New York City.
She addresses "methods of [attaining] freedom and recognition," as you mention, when writing about the "sobbing" school of Negrohood, a group from which she distinguishes herself. Hurston was referencing fellow writers in the Harlem Renaissance who focused on the more tragic elements of black identity. For Hurston, the flaw of this "school" lay in what she perceives as its pessimism, its persistent sense of injustice, and its backward stance toward life:

I do not belong to the sobbing school of Negrohood who hold that nature somehow has given them a lowdown dirty deal and whose feelings are all hurt about it. . . . No, I do not weep at the world—I am too busy sharpening my oyster knife.

The answer to attaining freedom was to perceive oneself as free. The world need not be a thing at which to "weep"; according to Hurston, it is a treat that, with some effort and even a bit of struggle, one could open and enjoy.
The fact that she is "the granddaughter of slaves" is no cause for "depression." For her, slavery was the price that she "paid for civilization" and access to the world. Conversely, she argues that her "white neighbor" has a more difficult time in life, for "no brown specter pulls up a chair beside me. . . . The game of keeping what one has is never so exciting as the game of getting." For her, it is an exciting time to be black. The whole world existed with "[everything] to be won and nothing to be lost." On the other hand, the "white neighbor" feared that black gain necessitated his loss.
Hurston had "no separate feeling about being an American citizen and colored"—and thus rejected W.E.B. Du Bois's notion of "double-consciousness." Instead, she indirectly references Walt Whitman's "Song of Myself" by describing herself as "merely a fragment of the Great Soul that surges within the boundaries." This sense of being a part of the universal collective gives her the sense of freedom to be herself ("When I set my hat at a certain angle and saunter down Seventh Avenue . . . feeling as snooty as the lions in front of the Forty-Second Street Library, for instance"). This sense of being "full of herself"—that is, aware and proud of all of her elements—leaves her astonished when someone discriminates against her: " How can any deny themselves the pleasure of my company? It's beyond me." That company includes not only her cultural and feminine expression, but also her extraordinary individual self, which, to Hurston, matters far more than anyone else's ideas of who she is.