It is difficult for historians to answer a question about the privately held mental belief of a historical figure with much confidence, especially so complex a figure as Lyndon Johnson.

That said, the best answer is probably "yes." Lyndon Johnson, as a product of his civilization and time, was very likely to have held opinions we would now characterize as racist. This does not, however, mean that he was not, during his term as President, a supporter of the Civil Rights movement and related causes. His "Great Society" programs were broad-based programs to, as he saw it, "uplift" the whole of the country, and a large factor of that was improving the lot of disenfranchised blacks.

It would be difficult to make a strong case that LBJ and MLK worked together on Civil Rights. Johnson continued the FBI surveillance and wiretapping of MLK, and in private was known to refer to him as a "hypocritical preacher," with some derision. Nevertheless, the success of the Selma actions were a large spur to LBJ's push for the 1965 Civil Rights bill.

Caro's work talks a lot about this and it is pretty clear that he was not a racist, per se, but felt paternalistic toward the great crowds of blacks that would flock toward him. He decried the use of the way southern politicians continuously used the fear of a rising black man to keep southern politics predictable. He was undoubtedly a product of his times who used the n word but it is my opinion that he was not a racist. My memory is that LBJ really did not like MLK but was forced by circumstance to cooperate occasionally with him. It is an exaggeration, i believe, to suggest that they worked 'closely' together. LBJ was an inveterate narcissist, so it is unlikely that he would have worked well with someone else who would thereupon get credit.