Rasputin's actual influence on the court was relatively negligible. He had some sway on the Tsarina, but Alexandra had limited power. One of the most salient evidences of this is that Rasputin always opposed Russia's entry into the war. 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. 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 dealing with the sexual degeneracy under the aegis of Rasputin were huge sellers in the post-February popular literature). Thus while Rasputin the man had very little influence, his image had greater sway.

source

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

The book Russian Roulette, by Giles Milton, is about the involvement of the British spy agencies operations in Russia, and has a chapter on Rasputin's assassination.

Since you're asking how he really died, I assume you know the tale of him eating poison, getting shot, and otherwise acting like the villain from some horror story.

That's because it was a horror story, concocted by one of the conspirators, to cover up the involvement of the British secret agent Oswald Rayner, who delivered the fatal bullet into Rasputin's head, after two other conspirators shot him in the liver and kidney.

The cover story might have held up to scrutiny, except his body was discovered and an autopsy performed, which found no trace of poison in his stomach, and a body which otherwise didn't conform to the story that Prince Yusupov had spun.

Feliks Yusupov, fun fact, had lured Rasputin to his house with the promise of allowing the monk the opportunity to have sex with his famously beautiful wife.

If you like stories about secret agents and how the British built their organization up, it's a really good book. The guy who started it all would test the mettle of possible recruits by stabbing a pen into his own leg, and seeing how they would react.