What is the historical context (time period) of "How It Feels to Be Colored Me"?

I assume that you are asking about the period in which the personal essay "How It Feels to Be Colored Me" was written. The story was published in 1928, in the midst of both Modernism and the Harlem Renaissance.
However, in the essay, Hurston chronicles what Black life meant to her from her childhood in Florida in the 1900s to her womanhood in the 1920s.
Though Hurston loved black people, she often saw herself as apart from some of its concerns and agendas, a sensibility that is evident in this essay. She writes of how she would come out onto her front porch to watch white Northerners go by. For the rest of the town, the front porch may have been "a daring place," but for her "it was a gallery seat." Hurston contrasts the fear other black people had toward whites with her curiosity, even her willingness to greet and speak to them.
She goes on to write that she "is not tragically colored" and does "not belong to the sobbing school of Negrohood." This places her in slight contrast, at least in tone, with other writers from the period, including W.E.B. DuBois who argued in The Souls of Black Folk that the condition of black people is like that of someone born covered in a "veil," an instrument that both obscures the identity of the person wearing it and allows them to see others in a distinct way.
Hurston also knows what a gift Blackness is, for it gives her an ability to appreciate black art forms in ways that she thinks eludes whites, such as when she notices the difference between her white companion's reaction to the music and her own: "He has only heard what I felt."