According to RFK's memoirs, when the 1964 Civil Rights Act was passed, LBJ boasted "We'll have those niggers voting Democratic for the next 200 years".
LBJ may have been a racist, but he recognized a political opportunity when he saw it.
Interesting to note, by the way, that LBJ even apologized for using the N word just five years later.
It's interesting that so far nobody has mentioned the Estes scandal that engulfed Johnson during Kennedy's first term, and the suppression of the investigation allegedly at the behest of the Oval Office.
The theory advanced by many, as documented in articles in the Chicago Tribune and other papers, was that Kennedy, while not liking Johnson personally, found him useful politically and was therefore willing to have the investigation slowed at least until after the 1964 elections to ensure his own re-election. Those same accounts mention that RFK didn't care for the idea (he despised Johnson) and that caused something of a rift in the Kennedy family.
Kennedy and Johnson did have a somewhat turbulent relationship, true, but not quite as bad as what Nixon would have with William Scranton later on in 1970-72, over the "Law and Order" riots and anti-Vietnam protests(so much so, that by the time that the GOP Convention was held, Scranton absolutely refused to run for another term as VP and we eventually got President Ford out of that deal).
Had JFK been assassinated, his brother could have shaken off the go-slow order and gone full bore after Johnson, quite possibly resulting in his impeachment. Had that happened, don't discount the number of old school Dixiecrats who would have voted for conviction out of sheer vindictiveness. The result would have been Johnson marginalized for the remainder of his tenure, or a John McCormack presidency.
It's quite possible that RFK might have gone for impeachment, but it doesn't seem likely at the same time. And then there's the fact that Johnson probably would have picked a VP straight away after Kennedy died, so Johnson's VP probably becomes President(maybe Hubert Humphrey?) in the event of an impeachment anyway.
I don't doubt that the Dixiecrats, being the hardline right-wingers most of them were, would have gladly screwed over LBJ, though: hell, after John F. Kennedy stepped in to stop the Chicago riots from blowing up(he later had to call in the National, IOTL a few of them actually wanted to impeach *him*!