I applied and received the same at the Russian Embassy in Bangkok. I'm a legal resident of Thailand.
The requirements are pretty straightforward:
a) Apply in US OR the country where you have legal residence status. (This will be verified by the local Embassy - I suggest putting a little page sticker in the Passport if it's at all full;
b) per . . . Business Visa | Embassy of the Russian Federation to the United States of America
- Written statement from the host organization, which should give: the full name of the host organization, its official address and contact information, the surname, first name, patronymic, and position of the person who signed the statement (if the host organization is a legal entity or enterprise, an institution, or their affiliate established in the territory of the Russian Federation - the individual taxpayer number), and basic information regarding the visa applicant: the surname, first name, other names, date of birth, citizenship, sex, passport number, multiplicity of the visa, purpose of travel, requested period of entry, location of intended residence of the visa applicant, and the cities to be visited.
If you have the above, great. If not, here's an example for your Russian host - http://www.visahq.com/pdf/Russia_3_year_bus.pdf
c) Money in USD.
d) Visa Photo
e) Visa application completed.
f) Other local requirements - Check the local Embassy's requirements. In Thailand a Health Certificate was additionally required.
The only issue I had . . . at the Mongolian/Russia crossing at Tsaaganuur/Tashanta . . . the Russian staff examined my Business Visa issued in Bangkok for a long time and made several calls before letting me continue.
I remember the Russian Embassy in Bishkek. Just up the street from that silly 'Sierra' coffee shop.
I was dreaming of Peets and found a watery Starbucks fern bar.