If you have a B.S. or B.A., or a higher degree from a university, it will be much easier to find work. With these, you can link up with a university that specializes in English. The pay in these is much better than other forms of teaching English as a foreign language.
Without that, your next best bet is to find a NGO (non-governmental organization) whose mission is to help and educate people in Russia. Most likely, a position with this type of company will land you in a rural area of Russia or smaller city. In my opinion, it is just as exciting as living in Moscow...
Another option, with a university education, is to work for the Peace Corp. Right now, they are operating in eastern Ukraine, which is primarily Russian speaking.
You can also teach privately, but you'll need to do some visa trickery to be able to stay for more than 30 days.
The most important part is to secure a job offer from someone, because if you don't, you won't be able to get a visa to work or a visa that is valid for a significant amount of time. Whatever you choose, good luck!