Robert Kennedy's dislike of LBJ dates back to before they even met (in 1953), and probably started with the latter's humiliating (for the Kennedy family) re-telling of the removal (in 1940) of RFK's father, Joseph Kennedy, as ambassador to the UK. Added to this were the many personal (especially background and personality) and political differences between LBJ and RFK, and other comments made by LBJ about the Kennedys which angered RFK.
Robert A. Caro's The Passage of Power goes into some detail on this, dating RFK's antipathy towards LBJ to before they had met for the first time. Leaving aside the political aspects (especially disagreements over Vietnam) which are mentioned in the Wikipedia page on Robert F. Kennedy, Caro relates the first meeting between the two in January 1953 in the Senate cafeteria where RFK was sitting at a table with Joseph McCarthy and others:
As Johnson, Busby and Reedy walked by, McCarthy, as was his custom,
jumped up to shake Johnson’s hand, calling him, as senators were
already starting to do, “Leader,” and McCarthy’s staffers also rose—
except, quite conspicuously, for Bobby, who sat unmoving, with a look
on his face that Busby described as “sort of a glower.”
Lyndon Johnson knew how to handle that situation. Moving around the
table, he extended his hand to take McCarthy’s and those of the
standing staffers, and, when he got to Bobby Kennedy, stood there,
with his hand not exactly extended but, in Busby’s words, “sort of
halfraised,” looking down at Kennedy. For a long moment Kennedy didn’t
move. The glower had deepened into something more. “Bobby could really
look hating,” Busby says, “and that was how he looked then. He didn’t
want to get up, but Johnson was kind of forcing him to,” and finally,
without looking Johnson in the eye, he stood up and shook his hand.
Later, after the Johnson group had finished their breakfast and were
leaving the cafeteria, Busby asked, “What was that all about in
there?” and Johnson replied, “It’s about Roosevelt and his
father.”...The long relationship between Joseph P. Kennedy and
Franklin Roosevelt had ended in acrimony and bitterness, and Johnson,
the young congressman, was a Roosevelt protégé.
To make things worse, LBJ had for many years been telling the story of how Roosevelt (FDR) had fired Joseph Kennedy as Ambassador to the UK (LBJ had been in the presence of Roosevelt when FDR had spoken to the elder Kennedy on the phone):
Johnson related how Roosevelt, in his booming voice, had said, “Joe,
how are ya? Been sittin’ here with Lyndon just thinkin’ about you, and
I want to talk to you, my son. Can’t wait.… Make it tonight,” and
then, hanging up the phone and turning to him, had said, with a smile,
“I’m gonna fire the sonofabitch.”
This was, unsurprisingly, humiliating to RFK, who was very close to his father, especially as Johnson, a skilled raconteur, told the story many times and, apparently, with great relish while making use of his gift for mimicry.
LBJ friend and advisor Tommy Corcoran related another incident when RFK felt the Kennedy name to have been slighted by LBJ, this one concerning the run-up to the 1956 presidential election and Johnson's unsuccessful attempt to gain the Democratic nomination. At a meeting with Corcoran, Joseph Kennedy, accompanied by his son RFK, made an offer:
If Johnson would announce his intention to run for President and
promise privately to take Jack Kennedy as his running mate, Joe would
arrange financing for the ticket.
Source: Robert Dallek, 'Lone Star Rising: Lyndon Johnson and His Times 1908–1960'
Corcoran put the proposition before Lyndon, who turned it down.
“Lyndon told me he wasn’t running and I told Joe,” Corcoran recalls.
“Young Bobby … was infuriated. He believed it was unforgivably
discourteous to turn down his father’s generous offer.”
Corcoran also relates that the calmer JFK was “more circumspect" about the rejected offer. Geoffrey Hodgson, in JFK and LBJ: The Last Two Great Presidents, also mentions a later incident of LBJ attacking Joseph Kennedy, this time at the Democratic National Convention in 1960. LBJ
attacked the founding father of the Kennedy clan.... Joseph P. Kennedy,
ambassador to London before World War II, had been distinctly
sympathetic to those British conservatives led by Neville Chamberlain,
the prime minister always portrayed by the cartoonists with an
umbrella, who sought to appease Hitler. “I wasn’t any
Chamberlain-umbrella policy man,” LBJ assured the Washington state
delegation. “I never thought Hitler was right.” Robert Kennedy, for
one, never forgot or forgave this slur on his father
But there was more to it than that:
But there was also the aspect that lay beyond the political, and
beyond analysis, too, the aspect that led George Reedy to ask, “Did
you ever see two dogs come into a room … ?” There was Bobby’s hatred
for liars, and his feeling that Lyndon Johnson “lies all the time …
lies even when he doesn’t have to lie.” There was his hatred for
yes-men—and for those who wanted to be surrounded by yes-men—and
Johnson’s insistence on being surrounded by such men, an insistence
which, Bobby was to say, “makes it very difficult, unless you want to
kiss his behind all the time.” He detested the politician’s false
bonhomie, and Johnson embodied that bonhomie. “He [Bobby] recoiled at
being touched,” and of course Lyndon Johnson was always touching and
hugging. And talking. “It was southwestern exaggeration against Yankee
understatement,” Arthur Schlesinger has written. “Robert Kennedy, in
the New England manner, liked people to keep their physical distance.
Johnson … was all over everybody.” So many of Bobby Kennedy’s pet
hates were embodied in Lyndon Johnson.
Caro also cites Harry McPerson, who was LBJ's counsel and speechwriter, and William vanden Heuvel, assistant to RFK:
Says Harry McPherson, who had worked for Johnson before the vice
presidency, “If your brother is President, and you’ve got this
powerhouse accustomed to being in command as Vice President, it would
make you as suspicious as anything.” Kennedy’s aide William vanden
Heuvel says that Robert Kennedy saw Johnson as “a manipulative force”
who could, if he ever got off his leash, be very difficult to rein in
again. So the leash had to be kept tight.
Mutual Contempt: Lyndon Johnson, Robert Kennedy, and the Feud that Defined a Decade by Jeff Shesol goes into great depth on the LBJ - RFK relationship. However, as Caro observes by citing RFK himself, there was a recognition of LBJ's consummate political skills:
“I can’t stand the bastard,” he once said to Richard Goodwin, “but
he’s the most formidable human being I’ve ever met.” “He just eats up
strong men,” he said on another occasion. “The fact is that he’s able
to eat people up, even people who are considered rather strong
Despite the animosity, LBJ and RFK could, and did, put on a show of mutual support in public. Johnson, for example, campaigned in New York for RFK when he contested a Republican-held New York senate seat (at the same time, many 'celebrity' democrats - among them Gore Vidal and Paul Newman - were supporting RFK's republican rival Kenneth Keating). In private, though, RFK's dislike of LBJ continued unabated.
RFK's New York senate campaign, 1964. Image source