Hi, all!
I'm Russian, - then my fav is Russia (of course). But...
Immediately after purchasing this game Russia wasn't any good but patch improved all slightly. Second step was after determining most favorable conditions for Russia. It is: Huge, Pangea, Warm, Wet, 5 bil yrs, 15 opps. In this conditions I usually play for Russia at Deity & win by Culture, Space or Diplomacy.
Little advice: never produce any Cossack (disrupting of their upgrade path is crippling!!!), try to avoid any war in Age after Ancient, Library/Univercity building rush is more important for you then acquiring big army.
Be open for any diplomatic activity after Ancient times (in Civ II all your spies was Vet - just for rememberance).
Repeat to yourself: "Best Russian wars was only - where Russia wasn't" (i.e. war for Greece independence, Afghan vs Britain, Chinese civil war, Vietnam etc.) then - behave accordingly.
Strange feeling - AI knows about high aggression level of Russia & respects you for that. Use that. Be Khrushchev beating your boot in UN assembly, but don't go to real war by no means!
Reminder: best Russian unit - its Scout. You must to find as many barb's village as possible & contact enemy Civilizations!
Take in your mind that you'll have only 16 cities as maximum due to corruption problem under Communism government (Communism was seriously crippled here in comparison with Civ II). Then - place them in "hive fashion" for taking as many place on chart as possible. Destroy any foreign Civ whose cities disturb your "hive". Destroy them completely for insurgency cause. Don't be gross for anybody else. It's just a business - nothing personal.
My second favorite Civ is - Iroquise. I prefer to use early "Expansion Dominance" then - their Expansionist treat for me is invaluable. Religious treat is must for Temple/Cathedral rush building. Mounted Warriors is nice extra. (This UU is really good in comparison with Cossacks, but Scientific treat has just been more valuable for me then Religious. Yes, Scientific is more important for me, then Religious & Mounted Warriors! Duh!)
Third favorite is - Aztecs. Their Jaguar Warrior is Scouts in disguise. (Little advice - produce them as many as usable, but never let them fight - you don't want your Golden Age in - Stone Age!) Religious trait is - never bad & has already explained. Military is - so-so, but usable for early Heroic Epic & culture dominance as result.
Fourth - Americans. Fifths - Englishmen. (I really addicted to Expansionist trait, but I don't like any wars - then no Zuluses

.)
Thnks & brgds, Alex.