It isn't a matter of whether HRC and Biden get along or not (and AFAIK, they do). The candidate chooses a VP candidate who adds something significant to the slate (i.e., will likely pull a block of voters who would not otherwise vote for the candidate).
JFK chose Johnson because he needed Texas's Electoral College votes if he was to have a chance at winning, and because LBJ brought along a significant number of votes that JFK couldn't otherwise pull, as well as uniting a significantly divided Democratic party.
Biden (much as I like him) doesn't add anything to Clinton's ticket - it would just be two relatively old white people. She's almost guaranteed to pick a Hispanic male, and one who is significantly younger than she. She won't pick a woman, because too many people are nervous about one woman, let alone two, on the ticket.
If Biden were to run and get the nomination, he would likely either pick a female VP or a Hispanic who is younger than he is.
I doubt that either would pick an African American running mate because (a) they will have the African American vote pretty much tied up anyway, and (b) too many people would have palpitations about another potential African American President so soon after the first one.