Doesn't seem to be fake, but as the anon noted, context is key.
It might have been when Biden was attacking William Bradford Reynolds in connection to some "New Orleans plan" or "Nunez plan" in 1982 (pics).
The N-word quote was reportedly made by Charles Emile Bruneau (he denied it later) talking to Lawrence E. Chehardy.
Book further says that Biden tried to use this to improve his position with blacks after he opposed to forced busing for racial integration.
(pics are from book: "Right Turn: William Bradford Reynolds, the Reagan Administration, and Black Civil Rights")