1. Anti-corruption unit in President's Administration (basicly responsible for identifying the corruption in federal government).
2. Prosecution office (reporting to President) - main anti-corruption body, general one.
3. Investigation Committee (reporting to President) - body responsible for higher-profile crimes than those of the Prosecution office, but basicly the more "elite" concurring agency.
4. Federal Security Bureau (FSB, reporting to President) - responsible for corruption cases that would pose danger to national security
If you wonder about bodies that would be able to prosecute against a President - there are none. President is immune from prosecution by the Constitution. It can only be initiated when he's out of the office (impeachment may be initiated by the Parlament though, but you have to gather 3/4 majority for that kind of thing).
P.S. About the video - it's all emotions and no evidence, the "blood-diamond-dealing warlord in sub-Saharan Africa" comparison may only sound feasible to someone who has no idea of what's going on in Russia. I could've said that it's part of anti-Russian propaganda, but I think the word is too pretentious, so I'll state that it's part of the common western information policy towards Russia, which does not change for like... 200 years already?