What kind of monk was Rasputin?

He was probably more a "starets" than a regular monk:

"The elder or "old man", known in Greek as geron and in Russian as starets, need not necessarily be old in years, but he is wise m his experience of divine truth, and blessed with the grace of "fatherhood in the spirit", with the charisma of guiding others on the Way. What he offers to his spiritual children is not primarily moral instruction or a rule of life, but a personal relationship. "A starets," says Dostoevsky, "is one who takes your soul, your will, into his soul and his will." Fr. Zachariah's disciples used to say about him, "It is as though he bore our hearts in his hands."

The starets is the man of inward peace, at whose side thousands can find salvation. The Holy Spirit has given him as the fruit of his prayer and self-denial, the gift of discernment or discrimination, enabling him to read the secrets of men's hearts; and so he answers, not only the questions that others put to him, but often the questions often much more fundamental - which they have not even thought of asking. Combined with the gift of discernment he possesses the gift of spiritual healing-the power to restore men's souls, and sometimes also their bodies. This spiritual healing he provides not only through his words of counsel but through his silence and his very presence. Important though the advice may be, far more important is his intercessory prayer. He makes his children whole by praying constantly for them, by identifying himself with them, by accepting their joys and sorrows as his own, by taking on his shoulders the burden of their guilt or anxiety. No one can be a starets if he does not pray insistently for others.

If the starets is a priest, usually his ministry of spiritual direction is closely linked with the sacrament of confession. But a starets in the full sense, as described by Dostoevsky or exemplified by Fr. Zachariah, is more than just a priest confessor. A starets in the full sense of the word cannot be appointed such by any superior authority. What happens is that the Holy Spirit speaking directly to the hearts of the Christian people, makes it plain that this or that person has been blessed by God with the grace to guide and heal others. The true starets is in this sense a prophetic figure, not an institutional official. While most commonly a priest-monk, he may also be a married parish priest, or else a lay monk not ordained to the priesthood, or even-but this is less frequent-a nun, or a lay man or woman living in the outside world. If the starets is not himself a priest, after listening to people's problems and offering counsel, he will frequently send them to a priest for sacramental confession and absolution."
Source and further information:
http://www.pigizois.gr/


2) "The Encyclopedia Britannica reports that at the age of 18, Rasputin went through a religious transition, eventually traveling to the monastery at Verkhoture. Here, he was introduced to the Khlysty sect. After traveling to the monastery and spending some time there, he did not become a monk. Even though he did not stay at the monastery to become a monk, this trip already set him on the path to power and fame."
Source and further information:
http://it.stlawu.edu/~rkreuzer/indv5/rasp.htm