I've never played Russia, but it looks decent enough to me.
Their UA means that you can build a lot more of certain units than your foes. Sure, everybody gets at least one horse in their vicinity, but iron is pretty scarce in my experience. IMHO this ability doesn't really become useful until the discovery of coal, which I can almost never find. Then oil, aluminium and of course uranium. In the late game Russia can become a nuclear superpower (debatably much like in real life). Even if you're not at war you can still profit from extra production and you can sell the resources to the highest bidder (except for uranium. I'm not experienced with late game wars but selling uranium to others seems to be something you only do when you have a deathwish or the profit you can make from it is simply outrageous).
The Krepost is pretty okay. It's one of the first buildings and functions like a normal barracks (which you want to build anyway if you're even a bit focused on military) and gives you a free expansion bonus. IMHO this makes half of America's UA obsolete.
The Cossacks... *sigh*. I love these guys in real life, but I believe that industrial cavalry is just so weak in Civ V that I doubt these guys are of any use. I think their ability fits them though, it encourages picking off isolated units, pillaging and getting out before they even know what hit them.
As for Catherine herself, I'm a bit in dubio about her. On one hand she was a very competent Russian ruler, responsible for a lot of Russia's expansion and bringing Russia closer to Europe in both geography (she conquered Poland) and culture (she brought the enlightenment to Russia). However, part of me believes she was merely put in the game because of her ... "interesting" personal life. What also bothers me is that she was pretty much turned into Civ V's eyecandy.
Seriously compare this:
to this:
Well, at least she looks respectable, unlike Catherine in some other Civ versions, because I can imagine that the real Russian noblewomen probably wore clothes not too different from her dress. I guess we can justify this by saying it's a younger Catherine that had just ascended the throne (I think she was in her 30's when she got her husband killed).
As for Russia as an AI player: HURGHH! I always see her on the other side of the continent as a stable but non-treathening civ. I always try to keep her on my good side and doubleteam another civ. She always backstabs me. Was Catherine this treacherous in real life?
EDIT: Damn, I got so carried away I forgot my conclusion.
Anyway, I wouldn't call Russia one of the stronger civs, but certainly not one of the weaker either. I'd put it on the higher half of the middle tier (though perhaps my opinion will change after giving Cathy a try).