Not so well. In Robert Caro's last LBJ volume he talks about the relationship in detail. LBJ saw JFK as a lightweight in the Senate. But JFK saw the value in having Johnson on the ticket because oil-rich Texas was a crucial fundraising state for Democrats. Once in the Whitehouse JFK basically ignored Johnson and essentially treated Bobby Kennedy as his Vice President. Caro writes that during the run up to the Cuban missile crisis JFK solicited Johnson's opinion and was horrified by what he considered Johnson's irresponsible hawkishness. Their relationship got so bad that many believed Johnson was going to be removed from the ticket in 1964 in favor of then Texas governor John Connally.

Not well.

Obviously, Robert Caro's multi-volume series is the ultimate history of LBJ's life and service. You should start there, and the fourth volume, which covers Johnson's interactions with Kennedy, has much of what you're looking for.

The relationship between Kennedy and Johnson stretches back to the U.S. Senate, when both worked together. Johnson had been in Congress since 1937 and in the Senate since 1949; Kennedy only started serving in 1953, following the '52 elections. "Now, this young man I appointed to the Foreign Relations Committee claims he knows more about foreign affairs than I do," Johnson said during the 1960 Democratic National Convention. (Caro, Passage of Power, p. 106)

That 1960 convention got heated between Kennedy and LBJ, who was also seeking the presidential nomination. Johnson raised "the Catholic issue" and Kennedy's health problems, while Kennedy's backers whispered about Johnson's 1955 heart attack.

John Kennedy himself appears to have a much more positive view of LBJ than almost any of Kennedy's supporters did ─ in particular Bobby Kennedy, who was LBJ's most implacable enemy in the White House. But John Kennedy was the one who picked Johnson as vice president to balance the ticket (Johnson was a sure bet to deliver Texas and a door into several other Southern states) and his campaign went along with it.

Once elected, however, Kennedy's staff and supporters ridiculed Johnson behind closed doors. These messages reached Johnson, who ─ particularly after his time in the Senate ─ had long ears. Bobby Kennedy was a particularly vocal foe of Johnson. Jeff Shesol's Mutual Contempt is a good book on their relationship.

Furthermore, Johnson was frustrated by the dead-end nature of the vice-presidency. He had given up a strong position in the Senate only to find himself sidelined by the Kennedy administration, which didn't really have a use for him after the election. LBJ wasn't one of Kennedy's close advisers; Bobby Kennedy was practically the shadow vice president.

In recorded interviews from 1964, Jacqueline Kennedy said her husband and Bobby Kennedy had discussed strategies for 1968 to prevent Johnson from running for president, and there's been ample discussion that Kennedy was seriously considering dumping Johnson from the ticket in 1964 in favor of someone like Texas governor John Connally. Kennedy's assassination changed all those plans.