Part of the series of pointers on getting a good fake Russian accent.
Rule 1. Know your cyrillic alphabet! (and your phonemes of Russian). The most basic way to try to get a foreign accent is to try to imagine the things you are saying being written in the orthography of the given foreign language. This isn't always effective, but it is a good starting point, because it's quite likely that, say, a Spanish person beginning to learn English, upon hearing an English word can mentally imagine it as a word written in Spanish that approximates its English sound. There are ways to improve on this significantly, but it all starts from here. So if you can read cyrillic, great! Here is the first sentence of this paragraph written in cyrillic.
Дэ мост бэйсик вэй ту трай ту гет а форин эксэнт из ту трай ту имэджин дэ тингз ю ар сэйинг биинг риттэн ин дэ ортографи ов дэ форин лэнгуэдж.
If you don't know cyrillic, but know IPA, it will look something like this:
/demost'bejsikvejtu'trajtuɡʲetə'forin'eksəntiztutraj.../ etc. etc.
If you don't know either cyrillic or IPA, well, then this is going to be a little tough - for now just try to read the IPA stream as if it were Spanish, except that /j/ is read as the 'y' in 'yes'. Learning the cyrillic alphabet isn't that difficult, though, and it is useful for much more than just pretending to have a Russian accent well! Get started!
You'll notice reading this that this is far from a perfect or even great Russian accent. Partly it is because I did the transcription on what I think is most reasonable from an English speaker's point of view (e.g. The - Дэ is not a transcription most Russians seem to follow, but if English is your first language, it's the one that I think makes the most sense). Partly it's because russophones speaking English realise that they're not actually speaking a gibberish language written in Russian. So there are problems. But already it's much better than most people can do. And what if you can't read cyrillic, or at least can't transcribe on the fly? I would suggest trying to practice writing snippets of English phrases in cyrillic until it comes somewhat naturally. But there is something you can do that takes much less effort and practice time - learn the phonology of Russian.
The easiest way is to familiarise yourself with what consonant phonemes English has that Russian doesn't and what the Russian phonemes that might be used to replace them are. So, here they are:
/h/ (the "h" in "hat"). Most russophones don't have /h/. Some do have /ɦ/ (the voiced version which is actually very close to /h/ - like the initial "h" in "huh" said by a stereotypical neanderthal) - but I haven't heard any of those people use it for /h/, probably because because of the vagaries of cyrillic orthography, the /ɦ/ is associated with /g/ and not /h/, and in any case, most russophones don't have this phoneme. Instead, the closest associated phoneme is, first and foremost /x/ (like the "ch" in scottish "loch"), for which /χ/ (like spanish "j") is an allophone for some speakers (e.g. me). In extreme cases, even /g/ pops up. You should generally go with /x/. If you are having trouble pronouncing it, try to start saying "k" and just draw it out. And then try to do the same without the onset "k" part.
/ɹ/ (the "r" in "rat"), or for that matter the rhotic vowels such as the "r" in "care". The standard Russian phoneme is /r/ (like Scottish "r"), but a short, restrained one, nothing extreme like the Spanish "double r" or anything. /ɾ/ is a sometimes-used allophone. Less common (sometimes considered a defect) is /ʁ/. Also sometimes used (and considered defects) are /ʀ/, /ɰ/, /w/ and even, possibly, /ɢ/. Many russophones have trouble with the letter "r" and there is even a famous monologue by a popular comedian (Zhvanetsky) where much of the humour comes from the fact that none of his characters can pronounce "r" properly, and they all screw it up in different ways. The standard practitioner of the Russian accent should go with /r/, and leave the other ones to experts and the very adventurous.
/ð/ (the "th" in "that") is totally foreign to Russian ears. Depending on the circumstance, preference or sheer dumb luck /z/ or /d/ or /v/ is used. Infuriatingly enough, mostly /z/.
/θ/ (the "th" in "think") ditto above. Replaced by /s/ (AAGHHH!), /t/ or /f/. The only Russians who will have the dentals are those who were taught English when they were very young. And if they still have a Russian accent, it means they were probably taught somewhere in the Soviet Union. And that probably means they were taught (a russified) RP. So the /θ/ and /ð/ are very breathy. If you want to pretend to be a Russian who was taught RP as a kid, hell, you are brave, for one thing, and you probably don't need this guide for another, and you prolly should be writing it, for a third.
/w/ (the "w" in "win"). Many russophones - and all southerners, belarussians and ukrainians have this sound. But most do not. It's usually perceived as a "defective" /v/. So you should in general use /v/ for /w/. Although /u/ is sometimes more reasonable. And, again, many russophones can pronounce /w/ very well - they just don't necessarily remember where the /w/ goes and where the /v/ in English words.
/d͡ʒ/ (the "g" in "gin") is always replaced by /dʐ/. The thing to keep in mind is that /dʐ/ is not used in native Russian words, and is not actually one phoneme, but just a normal "d" followed by a "zh" pronounced further back in the mouth (see below).
If you want to be very precise, Russian also doesn't have these exact phonemes, though it has very similar ones which can be paired unambiguously.
/ʃ/ (the "sh" in "shin") is replaced by a /ʂ/, same sound but with the tip of the tongue further back in the mouth.
/ʒ/ (the "s" in "leisure") ditto its devoiced pair above. The replacement phoneme is /ʐ/.
/t͡ʃ/ (the "ch" in "chin") is possibly not the exact same sound as the phoneme /t͡ɕ/, used in Russian, which again involves putting the tongue further back. I can't distinguish these in speaking or listening.
/ŋ/ (the "n" in "think") is replaced by /n/.
The bottom line:
Normal English phrase: What are you thinking about?
Cyrillicised: Вот ар ю синкинг эбаут?
In IPA: [votarjusinkinge'baut]
