I recently read that whilst both Nicholas and Alexandra were multilingual, the Tsarina’s grasp on Russian was limited. My understanding is that the royal couple conversed in English and that at court, they spoke French. So, given that Rasputin was from a peasant background and had received relatively little formal education, how did he communicate with the Tsarina?
Short answer: Tsarina Alexandra and Rasputin would have conversed in Russian.
It's a myth that the Tsarina had a limited grasp on the Russian language. The Tsarina's Russian was poor at first - as you'd expect from someone who'd been brought up in Germany - yet it improved over the years while living in Russia.
Rasputin was introduced to the Nicholas II and Alexandra in 1905. By then, Alexandra had lived in Russia for around a decade and could hold a conversation in Russian.
Alexandra also spoke Russian with her children, and they would send letters to her in Russian, as this 1914 letter from her son, the Tsarevich, shows.
While Alexandra preferred to speak in English or French with those around her, she would speak in Russian with those who knew no other language - such as some of the palace servants.
Rasputin's own grasp of Russian was also somewhat tenuous - he could of course speak it, but his letters reveal a rather childlike way of writing.
You might therefore speculate that Alexandra, who wasn't as confident speaking in Russian, would have felt more comfortable speaking with a peasant like Rasputin, given that any mistakes wouldn't so likely be picked up on.