From all historical accounts, which have not, to my knowledge, ever been disproved, Rasputin was a horrific, monstrous person. Considering the amount of unchecked, blatantly overlooked pedophilia that exists in the Church today, it is not difficult to believe that such a horrible person existed in the role of a monk; it seems entirely plausible.
Rasputin was a healer. He also advocated equal rights for the oppressed minorities and the peasants, and wanted the Tsar to be closer to the peasants. This was a threat to the aristocracy and their status, so they began a vicious campaign against him and those wild stories that have been widely dispersed and written about were largely rumors intended to discredit him. He was considered by the nobility to be a traitor because: he was anti-war at a time when Russia was pro-war; and he advocated equal rights for the severely oppressed Jews when anti-Semitism was government policy, and the laws confined most Jews to one area (the Pale of Settlement), denied them education and most occupations. In addition, they were often denied their very lives, as the military conducted raids on Jewish villages, torturing and slaughtering innocent families - all in line with government policy. He was not a monk, and never claimed to be. He was a 'strannik' - a spiritual pilgrim.
It depends on what this question defines as a "monster." I think Rasputin might have taken part in a lot of questionable and unthinkable actions, but I have my doubts about the claims that he was almost invincible. I think that he was able to build himself a very powerful image in many people's minds, and the legend continued to grow far after his death.
I guess I'm not even sure how this is a question that can be asked with any bit of seriousness. There is no reason to think that Rasputin was, as this question phrases is, a monster monk. Was he a weird guy? Of course, but that doesn't make him a monster monk.
Rasputin was not a monster monk, but he did have a strange effect on some members of the Romanov family. He seemed to have some powers that were somewhat difficult to explain. He did cause concerns for individuals that were close to the royal family which could have hastened his demise.
Rasputin was an individual with peculiar beliefs. When these beliefs clashed with the bourgeoisie, it caused the opinion of influential people to look at him unfavorably. For example, he was anti-war in a time where war was the answer to most problems. His lifestyle also help support his detractors who did not approve.