LBJ's bias toward racial equality was rooted in his early experiences in the Texas hill country. Given a chance, that bias emerged but only so long as it did not conflict with his political ambitions. Robert Caro's book, Master of the Senate, tells that story including how LBJ passed civil rights legislation in '57 and '64.
But passing the bill was only half of what had to be done. This article in the LA Times summarizes what Johnson's administration did to enforce the provisions of the Civil Rights Act.
Both the book and the article are good reads.
One of the consequences of Johnson's victory was the Democrats' loss of the South. The interactive map in this article traces the geographic distribution of congressional districts 1922-2012.