On the lower difficulties (King or lower) you can do fine even great avoiding war and getting culture instead. Spend gold on cultural city states and rushing culture buildings and production buildings. Build every wonder you can but especially Cristo, Oracle, and Stonehenge (try to have at least 1 in each city). Get Piety and Freedom. Tradition is recommended as well for the wonder production boost in capital. The other two policies can be whatever though I'd recommend two of Order, Patronage.
In fact it may even be advantageous to avoid wars beyond possibly an early Swordsman war. Hammers and gold spent on troops are hammers and gold not spent on culture. I know puppets contribute, but they don't focus on culture anymore so towards the late game they really don't add much to the victory and may even slow down if those armies are costing you culture in your main cities.
Your Social policy path should be as follows in this method (assuming you are not saving policies for later):
Tradition->Aristrocracy (for wonder boost)->Piety->Mandate of Heaven-> start filling out the tree until you unlock-> Freedom->Constitution->Free Speech->Finish filling Piety for two free policies->Your choice of filling out Freedom and Tradition vs Starting Patronage/Order. Depends on the situation. A large map with many city states, I'd suggest filling freedom and getting as many cultural CSs as possible. Order for the+5 production in each city is nice. On very small maps, replacing Patronage with Commerce may be worthwhile if you have lots of luxuries. +3 production and +1 happiness per luxury could end up giving you more.
The difficulty doing this with a small empire is going to be gold. Make sure to trade off excess luxuries and strategics with the AI whenever possible. With the small army you have, take barbs whenever possible. If there are no barbs and you don't have any concern of invaders, keep your army as small as possible. If you never declare war or do anything to piss off the AI, they should trade with you pretty consistently.
On the higher difficulties, this becomes nearly impossible. You miss out on wonders unless you are way ahead in tech. The only way to be way ahead in tech is to be big. But hey, higher difficulties should be nearly impossible.
If you don't want to be a warmonger, simply take out whichever AI declares war on you early. This is more along the lines of "defending against a lunatic" and thus still fits with being cultural. If you really don't want to take any cities at all, just keep your city total to 5 max and that is pushing it. You may need to burn some gold on research agreements if you have that small an empire though.
