Johnsondoes better in his early political career, reaching the Senate in the 1941 special election, and Truman, seekinga Southerner who is against Jim Crow, picks him as his VP in '48. Johnsons' not as powerful but still is good at running things from the VP chair, the way John Adams always wanted the VP. position to be. Truman allows it because he, unlike Kennedy, doesn't mind the powerful Johnson doing this becasue he's comfortable as the incumbent, and Johnson does bring votes. They clash on the firing of MacArthur, and this dooms his chances of getting Truman's support in '52, so the not goes to Stevenson. Johnson goes back into the Senate in the '54 election.
Johnson is well known enough, and vindicated enough by the fact Korea became a stalemate, that he has the inside track on the nomination once 1960 rolls around, the "LBJ was right" vote wins him the nomination, and he captures the White House in a long, bitter fight with Nixon.
During this fight, JFK - who was picked for VP 4 years earlier (he was considered OTL) is stinging from the defeat, but is willing to accept Johnson's call to be the VP nominee again. He feels that, unlike the fight with Ike, this is a fight where he can look like the "nice young boy who helps the teacher all the time while the ruffians square off in school." And, he does - the image idea works, and he is seen as one reason Johnson won the White House over the unknown Lodge and the rascal Nixon. (We'll leave aside the obvious jokes about how he acts real nice in class but what he's actually doing after the school dance behind the football stadium.
Kennedy is concerned about Johnson's domineering attitude, but he lost once and felt lucky enough to be picked for VP again, so he has no qualms about biding his time and waiting for 1968. Besides, Johnson was VP under Truman and the connection to the FDR-Truman years makes playing second fiddle seem not as harsh as it would. He does Johnson's bidding as VP in working with the Senate and just figures he'll succeed him just like Truman did FDR eventuallyand anything to Bobby that's confidential but leaks out is easily met with, "This is far less incriminating than what you did to Truman over the MacArthur decision." And, Johnsonr eluctantly agres and lets Kennedy have his private reservations about Johnson's power grabs.