Dedicating city to some task means that you build that aspect enhancing building as soon as you have it available (markets, banks, stock exchanges for money cities, etc.) and you appoint specialists only to that same task. You still need culture to adopt those all important policies even if you are not going for cultural victory. That means that some of your cities would be appointed to making money (if they have a lot of luxury resources nearby), some to making science (if they got jungle tiles), some to production (a lot of hills) to build wonders and units, some to make culture (none of the above advantages). Constantly develop your cities, prioritize them. Raze them and leave a capital as a puppet (annex it then you have some spare happiness). Now heal your army and go and get those AI cities. By that time you should have discovered mathematics and built a catapult or two. AI should send several waves of units and run out of them at some point. Don't lose any of your units, if necessary retreat to heal. Do not attack! Select a good defendable spot of land (might be a river valley or similar), preferably inside your borders for a friendly lands fighting bonus and faster healing and get them to come. Try to get other civs as your allies in this war for now. Once you have 5-6 units declare a war to your nearest neighbour. Discover horseback riding and build some 2-3 horsemen. Discover archery and build some 2-3 archers. You might be able to settle 2 cities or maybe 5 cities (depending on the map) before you start rubbing your shoulders with other civs.

Be nice to them, but don't forget that they are your rivals and sadly due to a crazy inherent AI warmongering its almost impossible to make a good friend till the end of the game in Civ5. On the other hand its way too easy and some people call it an exploit, so I don't usually do that. Declare a war, capture a worker and make peace all in the same turn. If you met a militaristic CS you can steal a worker from them (they are useless). At this point you just build settlers, workers and monuments in all your cities. Get Liberty SP branch and Citizenship SP next. Once you got a scout use it to discover map further and use your warrior to deal with barbarians near your cities. Settle the land to grab the luxury resources asap, but don't go too far from your capital. Scout the land using your initial warrior and try to not be delayed by any spotted barbarians. Develop technologies needed to get those resources. My winning initial build order is worker-settler-scout. That means if you are playing for English - build naval units, if you are playing for French - build a lot of small cities, if you are playing for Romans - build all that is needed in Rome, etc. The specific faction strategy is to get the most of the special ability. A strategy might be slightly different depending on which faction you are playing, but there are some basic rules which work for all of them. You need to apply a flawless strategy to get a win on those. You can also use continents or Pangaea maps. You should be able to get any kind of victory quite easily. Just expand all the time, build some naval units to fend off any aggressive AIs, grab those resources, trade and develop.

The easiest way (as Irony Man correctly pointed out) is setting the map type to archipelago, as AI is totally hopeless on water. Just I have to warn that its much more fun if you discover it yourself rather than get it from internet, as in that way it kind of ruins your game experience as it did to me to some level. And you don't even have to make some special settings for the game. Apparently there is a flawless strategy in civ5 and if you apply it correctly, winning on deity is rather easy.

Civ 5 deity ai bonuses