First of all change your difficulty to Warlord (the next above cheiftain) as playing on Cheiftain won't really teach you anything.
If you attach a save of your game, then we could tell you exactly what you are doing wrong and help you to correct it. Some general tips:
-Specialize a few of your cities
Have one or two focus on the production of settlers. This means that they should have a large food supply (built on/near flood plains, or wheat or other food source nearby). Have another few cities focus on production, ideall built on or near some hills to capitalize on mines in the area. This will create a lot of shields for units and wonders. Finally, a few more (or most of the rest) cities should focus on commerce to make up for the other specialized cities' loss (and in general to strengthen your economy).
-Don't build a ton of improvements for your cities.
Only build the ones that are essential. If your cities shares a border with a rival civ's border, build a temple to increase your culture and take some of his land. In your specialized cities, build a library/university in a large scientific city, a marketplace in your high-income cities, etc. Don't build a temple in cities that aren't on a border, don't build a library in a city with little or no science output and such to reduce the cost of maintainence. Defintely build a barracks for any city that will be producing military units. This helps a lot when it comes to fighting.
-Trade. A lot.
The more you trade, the more friends you make. Just make sure you don't get ripped off. Try buying a tech off of a civ (may go for 60 gold or so, depending on the tech) and then selling it to the rest of the civs for gold per turn. I think in general the AI trades its tech away pretty quickly. If you have a tech lead, hang on to it. Don't sell your tech. If you have excess luxuries, the AI loves those, so you can sell them at a steep price. Consider giving gifts every now and then just for the heck of it. I conquered a city, but it was too far away from my capital to manage (too much corruption), so I gave it to a nearby civ that was my friend.
-Starting the Game
In the opening moves, produce as many settlers and workers as you can (but don't forget to produce a military for protection). Don't worry about improvements for a while. You want to grab as much land as you can, and make sure to grab the fertile land.
-Don't automate your workers
The computer manages your workers pretty poorly. As a general rule of thumb, mine grasslands and hills, irrigate flood plains and any other square that has food. Build lots of roads to connect your cities.
Anyways, if you attach a saved game then I could be more specific with your case.
There is also a whole topic full of tips here:
http://forums.civfanatics.com/showthread.php?t=59026