From an earlier answer of mine

Rasputin was a controversial and enigmatic figure even during his own lifetime, with rumor and falsehoods circulating around him that both magnified his role in the Russian government and his influence with the royal family. The strong dislike for Rasputin stems from both his unlikely ascent into the inner circle of the Empress Alexandra and his behavior once ensconced within this social network.

Rasputin's actual influence on the court and beyond was relatively negligible. Rasputin had come to the attention of the imperial couple in the aftermath of the 1905 Revolution. This timing was rather important because the Revolution had underscored the importance of religion as one of the few venues left that elevated the autocracy and Rasputin gained access because he was one of a number of holy men the imperial house patronized. Rasputin had came to the attention of St. Petersburg society by virtue of Olga Lokhtina, a wife of one St. Petersburg official and a formidable hostess of one of the city's salons. Lokhtina claimed that Rasputin had cured her of her various ailments and asserted despite his peasant manners, he was a true man of the people and a holy man. Once inside Tsarskoye Selo, Rasputin had some sway on the Tsarina, but Alexandra had limited power. Her German background was the subject of much negative aspersions during the war and her connection to this mystic added further credence that she was plotting to undo the Romanovs from within. A good majority of the Russian court saw him as a disreputable peasant social climber who attracted an unsavory element within the court. Rasputin's drinking binges and sexually-charged parties were both real, but also grew in their retelling in Russian court gossip. Such gossip also cast Lokhtina, and sometimes even Alexandra (who in previous tales had been cast as a lesbian) as one of Rasputin's conquests. While it is highly questionable whether he slept with the Empress, he did manage to seduce a number of high-born women. His access to the Tsarina meant that some ambitious hangers-on did try to curry favor with him, whether through money or sexual favors. These bribes backfired and Rasputin was seriously injured by a female assassin on 29 July 1914, the day after the assassination of the Archduke Franz Ferdinand. Rasputin's would-be killer, Khioniya Guseva, was follower of Rasputin's one time patron and then current rival, the monk Iliodor. Rasputin's influence in the court meant that the royal family tasked him earlier to reign in Iliodor's critiques of the government, but this only heightened the two men's rivalry with Iliodor attacking Rasputin's immorality and venality.

Nicholas II did not like Rasputin personally, but humored him for his wife's sake and the fact that he seemed to have some control over Alexei's hemophilia. The Emperor also occasionally used Rasputin as an envoy, but Nicholas II tended to keep the holy man at arms' length. One of the most salient pieces of evidence for Rasputin's limited influence was that he always opposed Russia's entry into the war. He had always opposed war as a tool of state and had earlier pushed for Russian neutrality in the Balkan Wars, but in this case, he was much more in line with the geopolitical and domestic currents that favored Russian non-belligerence. Such a context was not present in 1914, and Russia went to war despite Rasputin's commitment to peace. That said, rumors of Rasputin's debauchery helped to sap the monarchy of its prestige and moral authority during a particularly acute time in Russian history. Right before and after his death, many came forward to claim that Rasputin really ruled Russia, but this was largely a combination of scapegoating and scandal mongering. Pamphlets detailing the sexual degeneracy of Alexandra and Rasputin were huge sellers in the post-February popular literature when much of the censorship apparatus was dismantled. Even though Rasputin was assassinated the year prior to 1917, the tales of what it took to allegedly kill the man (poison, shooting, and drowning) added to his mystique. Thus while Rasputin the man had very little influence, his image had greater sway.

The often colorized, bearded image of Rasputin, complete with fiery eyes and massive beard, along with embellished tales of his debauchery and resistance to death, have made him a somewhat iconic figure in popular culture. As such, the historical figure has merged with the mystical and supernatural. The man was a symbol used for the goth metal band Type O Negative's album Dead Again whose inside cover replicates Rasputin's autopsy information. Alternatively, on tv, Rasputin has been a vampire or the lover of a vampire. In the 1997 animated film Anastasia, he was a wizard, while in the Hellboy comics he is a warlock compatriot of the folkloric witch Baba Yaga and later a ghost. As this sample indicates, Rasputin is not exactly hated in contemporary culture, but rather has become a stock historical villain in supernatural fiction. These tropes co-opt preexisting images and details about Rasputin's life and graft elements of the supernatural to them, all while taking all the many rumors about him as a given.

But it is important to realize though that while Rasputin was a somewhat immoral figure, he did cultivate some of this mystique himself. Images like the 1906 photograph or others which showcase his Charlie Manson-esque eyes and unkempt beard were part of his self-presentation and public persona as a true holy man from the provinces as opposed to the effete and Westernized St. Petersburg high society. His behavior and dress also differentiated him from other Orthodox leaders, but the existence of rivals like Iliodor and even Father Gapon, who lead a worker's movement in the runup to the 1905 Revolution, shows that there were a number of Russian religious figures who used Christianity to gain a public role. By being such an anomalous figure, he stood out from many of his contemporaries who also curried court favor. Rasputin's success at gaining such favors was reason enough for a number of St. Petersburg elite to hate him, he need not be an immortal sorcerer vampire ghost, or even a mundane puppet-master of the Tsarina, to elicit such opprobrium among his contemporaries.

Sources

Figes, Orlando, and B. I. Kolonit︠s︡kiĭ. Interpreting the Russian Revolution: The Language and Symbols of 1917. New Haven, CT: Yale University Press, 1999.

Fuhrmann, Joseph T. Rasputin: The Untold Story. Hoboken, N.J.: John Wiley & Sons, 2013.