My time to shine.
I'm in a fairly unique position to judge this movie as I've been very close friends with LBJ's top aide during his second term for over a decade now. I've read huge amounts of material from their daily briefings, much of which still isn't approved for public release, and have fairly detailed non-public knowledge of LBJ from a first hand source.
I was at the premier screening last week in Atlanta. Aside from a few glaring exceptions the movie mostly got things correct.
He could absolutely be somewhat crude at times. The "leave me some room for my nut sack" is a direct quote from a phone call with an executive from the Haggar pants company. Talking to people on the toilet with the door open was also a regular occurrence. It would be a Sorkin style walk and talk except mid way through Johnson would stop to use the bathroom with the door open and act like nothing out of the ordinary was happening.
His method of handling Congressmen and Senators, what's known as "The Johnson Treatment", was definitely understated in the movie. The scene in the elevator was spot on. I actually have a set of the cuff links he would use to do this. I don't think they went into enough detail here. It was actually far more intense and effective than shown in the movie. The deal he reached Senator Dirksen on the Civil Right Act was only reached after LBJ brought him into the oval and had documents ready and awaiting his signature that would move a large VA hospital expansion out of Dirksen's district. This would have been a far more effective and entertaining example of the Johnson Treatment than a throwaway line saying "we reached a deal with Dirksen".
His relationship with MLK and the civil rights movement was reasonably accurate. Their relationship was closer and more collaborative than the movie let portrayed. The wiretaps 100% existed and were used to monitor King, and to a greater extent his associates, to ensure they didn't do anything to compromise the process of getting civil rights legislation passed. Regardless of your view of the morality of the wiretaps the undeniable fact is that through a deeply divided government with the help of a brilliant public campaign by King,LBJ passed the two most important pieces of civil rights legislation since the civil war.
The largest point of contention was his treatment of VP Humphrey. Johnson never treated Humphrey that poorly. He could certainly be tough but he was never downright cruel as shown in the film.