There are 33 letters in the Russian alphabet. Take a look at them here: http://www.geocities.com/colosseum/track/7635/alphabet.html
Familiarize yourself with the possible sounds that they make.

One of the best ways to fake a Russian accent is to picture the words your saying spelled with these letters, and if you have familiarized yourself with the letters enough then you will end up saying English words with Russian sounds….hence, a Russian accent.

Besides this approach, there are a few quick tricks. First, replace w’s with v’s. Replace th’s with t’s.

For vowels, elongate your vowel sounds. Replace soft/short i’s with long ee’s or long i’s. With short e’s, picture a y in front of the vowel so that you’re making a ‘yeh’ sound instead of an ‘eh’ sound.

And, most of all, good luck!