In my novel, the MC goes to a Russian mafia in Moscow. Now before the MC character introduces herself as an English speaking person, will the Russian character start the conversation in Russian or as the book is in English, they will speak in English. Also, how will the mafia talk with each other, English or Russian?
The answer to your question will depend on the viewpoint of the narration.
Is the story told from the perspective of a character who doesn't speak Russian?
Then the Russians should not be understood by the reader either.
As Russian passages will irritate the readers, and as your viewpoint character will be unable to parse the speech, I would not render the original Russian dialogue ("Привет, Женя. Как дела?"), but rather describe how the character experiences not understanding what is being said ("The bald man shouted something angrily.").
Is the story told from the perspective of a character who does speak Russian?
Then the Russians should speak in the language of the narration (English).
Remind the reader that the characters are speaking Russian every now and then ("... he said in Russian.").
You might pepper the dialogues with some typical brief Russian utterances such as "davaij" (hurry up and other meanings), "zdrazdvudje" (thank you), "spasiba" (thank you) and so on which aren't essential for understanding the discussion and/or can be guessed from the context. (Check for the correct transliteration of the Cyrillic into Latin, I just quickly typed these examples without checking.)
Is the story told from an omniscient perspective?
Then the Russians should speak in the language of the narration (English).
Again, tell the reader what language is spoken ("Bla bla, said the Russian in broken English. You can speak Russian with me, the American replied in fluent Russian.").
For the reader it is of utmost importance he understands the significance of the scene. Why is the MC in this spot, why is he talking to these people. Does the reader get a more clear picture if the reader understands what is being said? Or should the reader just understand how the MC feels and struggles as he tries to communicate with the Russian person?
Since you are writing in first person and from the past, the MC knows or should know what has partly transpired in the scene. And because the scene exists the reader should know also.
So if the reader needs understand what the Russian tries to say to the MC then you should write in the same language as your story. But obviously make the MC oblivious to what the Russian is trying to say.
My answer to your other question, here, should also answer this one. In brief, English is the language you're writing the novel in, so English is the language you're writing their dialogue in. English is the only language you can expect your readers to read.
If some characters are speaking in Russian, and your POV character doesn't understand what is being said, you can tell that people in the room are speaking in Russian, your MC doesn't understand, then he introduces himself and they switch to English (or have a translator, or something).
If your POV character does understand Russian, you can just have the dialogue in English. Remember, you are not telling "what happened", but the MC's perception of what happened.
If the POV character doesn't understand Russian, but what is being said is important to the story, maybe some other character can repeat it to him in English - whether just give him the cliff notes, or make it interesting with side comments.
Another element you should not forget is tone and body language. While being unable to understand what is being said, your character might grasp the gist of what's going on through how it's being said. He might even glean some interesting insights: turn off one sense to heighten others, as it were.
Google Translate is not the best way to handle the Russian Language in the scene (check out the Youtube Channel "Google Translate Sings" for a hilarious reason as to why).
If you absolutely recommend that if you want your characters to speak Russian to find a fluent speaker to both translate into Russian and Transiliterate that into Latin Alphabet (as opposed to the Cyrillic Alphabet used by Russian, which has 33 characters to Latin's 26. As a fun fact despite this, sounds mostly have a one to one aspect between Russian and Latin Alphabet Languages so a Russian word written in Latin script will sound about the same as if it was written in Cyrillic.).
This is important to because all languages are subject use certain idioms and expressions that don't translate well but can convey similar meanings to other language's idioms and expressions. Russian for example, has a whole series of jokes that are funny if you understand Russian Animal character sterotypes, but fall flat when translated because the cultural norm is not there ("The Bear, the Wolf, and the Fox are playing cards. The Wolf shuffles and warns the table 'No cheating. Anyone who cheats will have her smug face bashed in.'" Is not funny in English but is hilarious for reasons in Russian. In these stories, it's customary to refer to the character's gender as being the same as their grammatical gender (which doesn't exist in English... or rather does, but would we tend to use gender only when refering to things that can have a biological gender). Of the three characters, only one is a feminine in Russian (the Fox) and even then, the sterotypes still need to be understood. Basically the violent wolf is threatening a sneaky female fox because the only other person in the game, the Bear, is both a male character and simple minded. Funny in Russian, or with understanding. Dull in English. Don't worry, the funniest joke in the world (the one where Sherlock Holmes and Watson go camping) is of Russian origin...).
It's also important to understand if a character's translation needs to be made available for the audience but not the non-speaking characters. This again can be done in a number of ways. You can use the actual spoken Russian, which will tip off your bilingual readers prior to the translation to the non-Russian readers that the mafia is about to double cross the hero (see X-Files where in one episode Mulder and his hated former partner (who speaks Russian) are imprisoned in a Russian Prison. One of the other prisoners tells Mulder to be careful as the guards are using formal language with the former partner and not informal language they use when speaking to other prisoners (implying the partner is not giving an honest translation) and easily overlooked by Russian Speakers. Another fun fact about Russians is that since American films tend to have Russian villains cause cold war, but few Russian speaking actors, American films and literature tend to feature fake-Russians who make rather amusing mistakes when speaking. This requires a lot of leg work to find someone who can help take your needed dialog and translate it to something a Russian mobster would say.
Another example is to go the comic book route where all languages are rendered in English, but certain other visual cues will denote that someone is being translated (usually they will use instead of quotation marks and add a note of what the actual language is being spoken. Star Trek uses this as cannon holds that most alien characters (and possible some humans) are speaking in their native language and are understood as English because they have "Translator Chips" that can do a really efficient translation. There are rare breaks, where a bit of Klingon language may get slipped in because there is no English equivelent, or in one case where they stopped working properly (the scene would swap perspective so the viewer was always able to hear in English, but only one side of the conversation) and in one case, worked find but the species encountered used so many cultural references it was effectively useless (which is a problem for language translators. I recall listening to one skilled Japanese-English interpretor discuss frustration because the nearest and best direct translation into Japanese doesn't convey the crassness of Donald Trump's infamous English statement "Grab'em by the Pussy".). Though a better example of this in television is the British Comedy Series "'Allo 'Allo" which was about the hijinks of a group of people in the French Resistance. While all actors spoke English, much of the humor was the result of Language issues being based on English accents. The French characters would use a sterotypical French-English Accent when speaking French, Germans would use a German accent to represent German Language, and the British all used over the top British Accents to convey speaking English. This allowed multiple scenes where characters would switch accents to try and speak the correct language to each other. One regular character was a British Spy working under cover to aid the French-Resistence had a quirk where he spoke poor French, in show represented by him speaking in a French accent, but loaded with malapropers (i.e. His signature greeting was "Good Moaning" instead of the correct "Good Morning". As the series continued, any accented utterance of "Good Moaning" became a tip off that the speaker has swapped languages and is bad at speaking the new one.).
Another idea is that the character may speak English but pepper his statements with well known Russian phrases that English Speakers will understand. This is often seen as a little lazy by audiences, but these days these have evolved to allow swear words to get into works and not trip censorship rules for that particular audience. Having a Russian speaker refer to a female hero as a "suka" could be a great way to have a character call the heroine a "whore" when the Englissh line is a bit to risque. And this gag works without a second language as Engilsh is infamous for a sort of internal language barrier depending on if your an American or Brit (Canada never gets commented on but tends to use American terms but British Spelling). Both sides are aware of idioms of the other but may invoke them in ways that make the American and British characters cringe. For example, a Brit asking an American if he can "bum a fag" is amusing to both sides of the pond because the double meaning is well known to both. In Britain, writers will often show American women horrified at a man who asks "what time should I knock you up tomorrow?". The American Woman will here "what time should I impregnate you" while the British man is asking the innocent question of "What time should I wake you up (by knocking on the door)". And when the first Harry Potter movie hit theaters, British and American Parents took issue with 11 year old Rupert Gint saying the line "Bloody Hell" but which word was offensive depends on the country. The Brits were concerned about "Bloody", which is a lot more offensive, while Americans see the word as Rupert Gint being an adorable Brit (and had a little shock from "Hell" which the Brits were fine with, so long as it is not Bloody). Buffy the Vampire Slayer got a great joke on Americans as the credits featured a British character flashing a reverse V to the camera, which most Americans didn't know was essentially flipping the bird to the view every week. British fans were shocked the scene was let in at all, no less the credits.