As for it's accuracy, it gets the basic idea across pretty well. Rasputin certainly did have many lovers, though it's a matter of debate as to whether the tsaritsa was actually one of them. In describing the monk's murder, the lyrics actually understate how difficult he was to kill.
The main problem is that omits the main reason that Rasputin became so influential. The tsarevich, Alexis, heir to the throne, suffered from haemophilia. Rasputin was a mystic healer and it was believed - particularity by the tsaritsa - that he was able to at least partially cure the boy. It was this, rather than his sexuality, that gave him such power in Russia.