While the letter is anonymous (in the sense that it's not signed and it's not on any letterhead), a complete copy was found in J. Edgar Hoover's files and a congressional investigation verified that it was sourced from the FBI. Whether Hoover himself wrote it is apparently unclear, but he almost certainly knew about it, given where the unredacted letter was found. The phrasing of the letter does not directly call for King to commit suicide, but it ends saying in part, "King, there is only one thing left for you to do. You know what it is." King believed that it called for his suicide, and many others have agreed.

His death is a different story. Blaming an assassination or even an attempt on the US government has a long and storied history. Some of these are accurate (Castro was long a target of the CIA), some are not (evidence strongly ties JFK's death to Oswald), and some are in between (Salvador Allende almost certainly committed suicide in the midst of a coup that was probably backed by the CIA).

But the evidence that it was carried out by the US government is basically nonexistent, and those comments don't provide any evidence besides the equivalent of a knowing nod. King had a lot of enemies, including a large portion of the white population but also some in the black population who disagreed with his nonviolent ways. James Earl Ray was convicted for King's death based on reasonably strong evidence, but over the years, various claims have been made that King was killed by others for various reasons, mostly having to do with racism or anti-communism, carried out by the Mafia, the KKK, or some other loner at least as often as the government is claimed to be involved. To my knowledge, the claims are backed by little or no physical evidence and mostly involve people long dead who cannot confirm or deny the claims.

There's probably more doubt about James Earl Ray than there is about Oswald, but not a lot more.