I presume you mean a revolution stemming from a failed Civil Rights movement? That seems to be the most likely avenue for something like that to happen.
However, in my opinion, at worst you would get a few radical groups sniping at policemen and occasionally bombing cars. Of course, a full-on black insurgency might be in the cards in TTL's 1970s if you start such fun in the 60s.
Other than that there isn't much chance of a revolution; the hippies and their ilk were in the minority in the 60s and weren't especially violent to begin with.