Langston Hughes and Claude McKay were male African American poets who were highly productive during the period known today as the Harlem Renaissance. Both poets were pulled from elsewhere to Harlem, a section of New York City that by the early 1920s had been dubbed the “mecca of the New Negro.” Most of their obvious similarities may end there, as their poems tend to be very different in terms of form and language.
The form of Hughes’ best known poetry can be characterized as “open” (or “free”), whereas the form of McKay’s best known poetry can be characterized as “closed” (or "fixed"). In other words, Hughes’ poems tend to have lines of varying length and only occasional or irregular rhyming, whereas McKay’s poems are often modeled very closely on the English and Italian sonnets, with measured lines, predetermined length, and fixed rhyme schemes. Similarly, but less certainly, the language of Hughes tends to be more familiar and informal than the language of McKay. (I write “less certainly” here because, in looking at the poems side-by-side, I see a few works by Hughes, such as “The Weary Blues,” that use vocabulary that is at least as formal as that found in McKay’s poetry.)
The most challenging part of your question for me is the third part: “Compare/contrast how these reflect Du Bois’s idea of 'double-consciousness.'" Editors at eNotes are generally supposed to answer only one question at a time, so I will simply point you in a direction that may be helpful...
W.E.B. Du Bois famously formulated the idea of “double consciousness” in the chapter "Of Our Spiritual Strivings" in his volume The Souls of Black Folk (1903). Here, he writes:
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 twoness, – an American, a Negro; two warring 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. ...
Perhaps by looking closely at selected free-verse poems by Hughes (such as “I, Too” and “America”) and selected sonnets by McKay (such as “The White House” and “The Negro’s Friend”), you will be able to identify how works by these two poets exemplify what Du Bois has called (in language that today may sound old-fashionably sexist) the struggle “to attain self-conscious manhood, to merge his double self into a better and truer self.”