The closest thing possible would be to have King assassinated in 1958 by Izola Curry, a mentally ill woman. He was almost killed after she stabbed him in the chest in OTL. His death prevents some successes in the civil rights movement. Then, you have his absence allow the Nation of Islam and other extremist groups to fill the void created. You then have "Nonviolence" replaced with "any means necessary" as the dominant belief. If you have the perfect storm of Kennedy never being shot in Dallas and Civil Rights Legislation being halted, you'd have even more eventually move over to this camp.
This in and of itself isn't likely to set off anything massive though. You'd need a long decade of assassinations and violent events for moods to sway far enough for mass fighting. However, if you get the volatile mix right, you could see riots by 1967-68. For as bad as OTL 1968 was, you'd need it to be even worse ITTL.
No matter how you get it to that point, its close to impossible to have a traditional Civil War come out of the situation. You wouldn't see a CSA V. USA situation arise, but instead a cultural uprising in the sense that one ideology is going to fight against another (maybe even more) openly in the streets throughout the country. There isn't likely to be any cohesion between the "rebels" either. It would probably look more akin to the riots after the slaying of Martin Luther King in OTL cranked up past 11 with a dash of the Watts and LA Riots. You would also see some coming out in the streets to "put down" the troublemakers on their own only to make the situation worse off.
The response to such a situation could also make things better or worse and it is depending on who is President and when exactly such an event would occur. you could see results that succeed like Little Rock or you could see mass Kent State like disasters. To make a situation this volatile to make things happen to begin with would also make the paths unpredictable once you reach that point.
So that I'm not just saying, "ASB" or "Its impossible", I would describe this situation as one where you could easily get mass fighting between some people, but its not one that enough people will openly take up arms to split a house in two over. More likely, they'll burn the furniture and fuck up the wallpaper until the parents show up and calm the situation down over the course of a few months. I wouldn't expect the incumbent President to have a snowball's chance in hell in the next election.
In short, Civil War, No. Anarchy for a few months, yes.