How does Barack Obama's description of "slip[ing] back and forth" between his black and white worlds in Dreams from My Father relate to W. E. B Dubois's double consciousness?

First, let's look at what Du Bois means when he uses the term "double consciousness." Here's what he writes in The Souls of Black Folk:

The Negro is a sort of seventh son, born with a veil, and gifted with second-sight in this American world—a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one's self through the eyes of others, of measuring one's soul by the tape of a world that looks on in amused contempt and pity. One ever feels his two-ness—an American, a Negro; two souls, two thoughts, two unreconciled strivings; two warring ideals in one dark body, whose dogged strength alone keeps it from being torn asunder. The history of the American Negro is the history of this strife—this longing to attain self-conscious manhood, to merge his double self into a better and truer self. In this merging he wishes neither of the older selves to be lost. He would not Africanize America, for America has too much to teach the world and Africa. He would not bleach his Negro soul in a flood of white Americanism, for he knows that Negro blood has a message for the world. He simply wishes to make it possible for a man to be both a Negro and an American, without being cursed and spit upon by his fellows, without having the doors of Opportunity closed roughly in his face.

In Dreams from My Father, Obama writes:

I learned to slip back and forth between my black and white worlds, understanding that each possessed its own language and customs and structures of meaning, convinced that with a bit of translation on my part the two worlds would eventually cohere.

Obama's longing for the worlds to cohere is articulated in the "two-ness" that Du Bois describes. Both men are tapping into the legacy that slavery has left in American society, even though the institution itself has been dismantled. That legacy creates a fissure that is, for Du Bois and Obama, unable to ever really disappear and that serves as the dividing line between the "black and white worlds" Obama describes.
Both men see being a black American as having to constantly negotiate this "two-ness" and feeling unable to truly collapse the words "black" and "American" into a single identity. This is what Du Bois means when he says that a black American feels "this longing to attain self-conscious manhood, to merge his double self into a better and truer self." Because America has such deeply racist roots, its social structures often prevent this "better and truer self" from ever developing because the double self has to divide itself into the white and black worlds Obama defines.
How very sad that something Du Bois was wrestling with in 1903 continues to be a lived experience for people of color in America today. When Obama was president, he often spoke about not seeing a liberal America and a conservative America but simply "the United States of America." May we all work harder so that there is not a "white world" and a "black world" but simply a world we are privileged to inhabit and thus are responsible for bettering and protecting.