And were supposed to believe this?
Hogwash.
Agreed. There may have been a real attempt at blackmail but a letter demanding suicide? Not even worthy of trash fiction.
MLK did receive such a letter. It was sent anonymously, so there is no irrefutable evidence that it came from the FBI, although they were (and are) the prime suspect since the letter was accompanied by audio tapes made by the FBI as part of their surveillance of King.
The letter did not explicitly demand that Rev. King commit suicide. It was an insulting litany of his “crimes” and misdeeds, and it closes with something along the lines of “there is only one way out for you” (which MLK himself, and many others, interpret as encouraging him to kill himself).
There are photographs of the letter posted out there on the web in various places.