It's a lengthy process and honestly not worth it. Even those actually in the Russian military don't necessarily want to be. They were just drafted. Conscription still exists in Russia.
Anyways, back to the original question. First off, you have to be able to read and write Russian fluently. You have to pass a Russian proficiency test. Second, you have to obtain a visa and then citizenship, I think. You might not be able to get by with a visa and have to wait 5 years to apply for citizenship (since I'm assuming you weren't born in Russia in the first place). Fortunately getting a visa is easier now, it used to be a bitch. Finally, you have to obtain a waiver from the Russian government (good luck).
Overall, it's not worth it. I'm pretty sure that you would have to become a citizen first before registering with the army. Also, Russia's not that great of a place at the moment. It's great for touring, but terrible for living. Although it is slowly making improvements, it still has a long ways to go.
Source: My mother was from Russia (ergo half of my family is from Russia), I had family members in the Red Army, and I've been to Russia several times.
