There are two main early warmongering strategies (I think): the first is to get access to iron quickly and rush swordsmen, the second is to get 'early' access to knights (works around turn 80-90 for best effect). Snarzberry will explain it better than me - he has a thread in this forum dedicated to it - but here are the things you need to do:
Technology: research pottery>writing and animal husbandry first. Your aim is to research Chivalry early, which allows you to build Knights. Spend as little time as you can researching technologies that do not lead to Chivalry. Once you get Writing you can sell open borders to AI civs, netting anywhere from 9-50 gold.
Wonders: if you can, build the Great Library. It gives you a free technology (get philosophy or, if lucky, Civil Service with it). If you cannot build the Great Library, try to build the National College in your capital. This is a national wonder - so each civilization can build it - and will boost your science output by enough to get to Chivalry quickly, but you'll need to have a library in both of your cities (build the one in the capital, buy the one in the second city).
Resources: Horses are an obvious necessity. You should try to have access to six of them, four is the minimum needed for this strategy. Luxury resources will be your primary means of getting money to pay for the unit upgrades; unless you are lucky and start next to both elephants and furs (those luxuries require camps to access, which are part of the 'chivalry tree', you will need to research either Mining, Calendar, or Masonry to improve your luxuries. Once they are improved, sell them to AI civs. The money you get for selling them depends on what each AI thinks of you; a friendlier AI will give you more money (up to 240 gold).
Social Policies: Take the Liberty tree; get Collective Rule after that. You are doing well if you can finish the entire Liberty tree by turn 80, as it will give you a free Great Person (choose Scientist, use it to research Chivalry).
Units: Build a total of four-6 horsemen/chariots. When you research Chivalry, upgrade them to Knights (or their unique equivalent).
City-States: find a cultural one, somehow get 250 gold, and then spend it to ally with that cultural city-state. This will give you the culture necessary to access the entire Liberty tree.
Cities: you need two, your capital and a second city. The second city must be built nearby horses, and you will need a worker to build a pasture over those horses to connect them into your trade network. A policy from the Liberty tree will give you the settler needed for your second city; waste no time in the capital building one.
A final note: explore and find as many civs as you can so that you have more people to trade resources/open borders to. Don't trade open borders for open borders, as some might want; also you might find it necessary to sell open borders and 1 gold per turn for 30 turns for 22 gold. This sucks, but if it gets you another knight it is worthwhile. Once you have upgraded all of your horsemen/chariots to knights, move them to just beyond an AI's borders and declare war on that AI. Crush their armies (avoiding pikemen!) and take their capital, maybe other desirable cities. Try to leave beaten enemies with one crappy city.
I think that's everything. Hopefully not too confusing?
You don't need to get knights by turn 80-90, but this strategy is most effective when you do. Similarly, you don't need a civilization that has a knight unique unit for the strategy, but those are the most effective.
I think that the total gold you'll need is 1130 (upgrading six horsemen (80 each), buying one library (400), and paying 250 gold to ally with a city-state). You can get this by selling two luxuries once each (480 gold) to friendly AI's, Open Borders (300) to six friendly AI's, discovering 4 city-states first or eight after someone else (120 gold) (or any combination thereof), and having an income surplus of 10 gold/turn for 23 turns. This is all perfectly achievable and can be done in a myriad of different ways, but the best way is to play on a standard or larger pangaea map with at least seven other civilizations. I'd suggest trying to get four knights and the Great Library to save 560$ from the maximum cost.
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You can also rush archers, which is a strategy best done with Babylon or Mongolia (DLC civilizations). Basically research Archery first, take the Honor tree and get the policy that gives you a free Great General (+20% combat strength for adjacent friendly units), hard-build 4-ish archers and buy ~2 more, then send that army plus your starting warrior at an enemy capital. Use the archers to kill the enemy army and take the capital's HP down to 0, then take it with the warrior. Babylon is good because they have an extremely powerful archer unique unit, Mongolia because they have an extremely powerful Great General unique unit. Regardless of which civ you use, this strategy can be used to kill off 2-3 enemies on a normal run.
Good luck with your warmongering!
