I always find myself behind in EVERYTHING when i try for domination. If i build more units i run out of money while the AI has this giant ass army 20 Turns in. Are you supposed to wait before you start an army or instantly go hostile when you get Catapults
The following will only work against the AI, or rookie human players. Most humans will spot your tactics and counter attack but against the AI it is mostly flawless.
Open the game as you normally would and play as if you were peaceful. Focus on growing and science, the two are related so fulfilling one will fulfill the other.
Your goals are composite bowmen and spearmen.
Composite bowmen are fierce at the time you unlock them and the faster you get them the longer their glory period lasts. You only need 2, for your first try I recommend 4 but remember that promotions make good units, godly and the more units you have the more those promotions get stretched outwards, delaying them.
A single spearman is fine but 2 again is probably better incase of a hiccup. The spearman moves into the city range triggering war if you haven't already and then you end turn to let it get shot. The AI usually (but not always) fixates on 1 target and that will be the spearman so leave him healing himself and his promotions should focus either on ranged defense or faster healing. Eventually you want as many promotions on him as you can get. Meanwhile move your bowmen in range and they should do around 20~ damage each per turn. A city heals 25 damage per turn regardless so eventually they will win the battle of attrition.
Alternately, you can replace the bowmen with horse archers and/or the spearmen with horsemen for incredible mobility. I personally use comp~ bowmen & spearmen/swordsmen because bowmen upgrade to crossbowmen but horse archers do not promote to other archer type units so whereas the bowmen will be useful all game the horse archers will not be. Spearmen eventually promote to a mounted unit anyway but spearmen is on the tech path of mining so it can help to pick it up afterwards unlike horsemen.
Comp~ bowmen require gods & kings expansion. Furthermore catapults are generally really bad; if only they got a buff or the setup requirement removed.
--Edit
AI has cheats on, trying to copy the AI will never work. A small army won't bankrupt you, a big one early in the game, will. Try to bribe AI to war each other so you can sneak attack the losing side. Raze rubbish enemy cities, or puppet them as this will force them to gold focus only which will help fund your war effort.