Sorry for the late reply. I'll try to summarize as much as I remember how I won the game.
First, I have to admit that I sinned a little bit - I disabled tech trading/brokering from the game settings. After reading comments to this thread
http://forums.civfanatics.com/showthread.php?t=479955,
I learned that disabled tech trade makes the game easier than it would be otherwise.
OK, as I wrote earlier, I rushed France and America with warriors. Some more experienced players may comment, but I guess that early military expansion contributed a lot to the victory.
However, by 990 BC my economy was crippled. I followed the advice about assigning scientists in my cities which helped me to keep my research going. I completed the Mids and adopted Rep and Slavery. Since my economy couldn't afford neither peaceful nor military expansion, I spent the whole classical era and most of the medieval era by developing my existing cities. My military was busy with resisting Barbarian attacks, but I captured a few of their cities too.
My most important trading partner was Charlemagne whom I tried to keep pleased/friendly. Somewhere in 8th-9th century AD, I agreed to assist Charlie in his war against Hammurabi. My knights easily ran over Babylonia.
The next target to my knights was Churchill. Because he is Protective leader, I needed trebuchets to support my knights on city attack. England, being on the verge of destruction, got vassalized by Bismarck who also was a master of Ramesses II. So, I suddenly was at war with 3 civs. Only to be soon followed by Sitting Bull who declared his own independent war against me.
My knights and pikemen easily resisted all attacks, but only after I got cuirassiers I managed to achieve decisive success on the offensive. In early 15th century I discovered Rifling and since then the AI was no match to my cavalry armies.
So, as a conclusion, my military strategy relied on mounted units starting from knights. Assisted by trebuchets/cannon while attacking cities and pikemen/musketmen/riflemen as stack defenders. Paris became my military powerhouse (Heroic Epic + 4-5 Great Generals) which constantly spawned Combat IV cavalry by the end-game.
There is a game control feature, I found extremely useful - ALT+left click on an unit production icon in city screen. Unit production queue is invaluable tool that helps to stay focused. No more dilemmas like "I have trained a Rifleman, what should I do next? Oh I know, let's build a Theatre! It would be way cool!"
2 years of civ4 experience and I didn't know before that such feature existed!
