Well, that's some for real weird shit. Can you imagine, black motherfucker becoming a legitimate (in a way) member of a KKK, with license and all? So yeah, apparently, that happened.
So, thankfully, my usual place has some nice info on what's true here and what's not. Apparently, the core is very much true - first ever black police officer decided to try for a KKK ad in a newspaper and just happened to establish contact with the organization. Motherfucker even used his real name, because he was sure nothing proper would happen. But it did, so for face to face meeting he had to send other cop, who's real identity is still unknown. Together they were the member of KKK, with a license, for 9 months, and even were almost made a leader of the local KKK chapter. Which, honestly, is hilarious on many levels.
Now, that, and some minor details represented in the movie, is true. However, most of the movie is actually full of dramatizations for the screen. There was no bomb plot, there was no storyline with that black power girl, we don't even know of the white guy was Jewish or not, cause, as I said, his identity remains secret. And the whole movie is structured to be very... how to say it... racial? It shows the clash of black and whites in US at the time, in a very focused way.
There are obvious reasons why it was done - race is a hot topic even today, apparently, for US, and without all those storylines the movie, while representing interesting factual event, actually would have been quite boring. Just some undercover police work, yep. Still, the event is based on is pretty damn funny.