Hi,
There are a few different Civs that are strong in economy, but one civ that I like to use and that is very fun to try out winning an econimic victory with is the Japanese. Simply put, you should build your city on mainland a stronghold and large city, then build all your other cities on small islands by putting settlers on ships. I typically put the first free settler on a ship and then build one new settler in all the new cities until I have about 5-10 cities depending on how many good locations I find. Switch the majority, if not all, cities to gold production.
The logic is as follows. On small islands you have many sea squares and they produce trade. The Japanese, get food from sea squares even before they have a harbor which make their cities grow even without any land squares. Finally, small islands are easy to defend and rarely attacked so you can focus on growth and gold instead of war/defence.
It is crucial though that you build your capital strong with alot of defence and culture to expand your borders in that area, otherwise you will be flooded by AI enemies. Another crucial part of this strategy is to be very active in diplomacy to buy your techs instead of researching them.
Here is a link to a very interesting thread discussing this topic. I hope you enjoy this playstyle because it is actually lots of fun.
http://forums.civfanatics.com/showthread.php?t=287242