I built a battery powered tesla coil myself.
One of the hardest parts of working with batteries is their life span. The primary circuit of a tesla coil draws a LOT of current. Batteries are rated in amp-hours, meaning the number of hours they can go if the load is drawing 1 Amp of current. A good battery has about 4-6 amp hours. However, a tesla coil absolutely has to draw more current than 1 amp in order to get anywhere near a respectable current going through the primary current (remember that the primary coil has high current, and the secondary has high voltage). So the batteries will die very quickly.
I built my coil using lantern batteries, 2 of them in series (12V), running through a pulse circuit and a car ignition coil to produce high voltage. The ignition coil will output somewhere around a 1-inch spark at a very high frequency.
The ignition coil method (which is by far, the most efficient tesla coil you can make with a limited power supply like batteries) works very well. I have used the ignition coil tesla coil method run plasma globes, make Kirlian photos, etc.
This is the way that it works:
An ignition coil requires a pulsed AC signal in order to continuously output a spark like a tesla coil. Batteries are DC, so you need to find a way to pulse the DC signal. Specifically, you need to make a square wave(a square wave is where it goes on-off-on-off thousands of times per second).
There are two ways to do this:
1. Use a relay wired as and electromechanical buzzer. It sounds very diificult, but it is not. It is $2.00 worth of parts and can be assembeled without soldering in less than 5 minutes. This way is the most similar way to the one that Tesla used.
2. Make an integrated circuit. There are a lot of people out there on the web who have used transistors to develop a complicated circuit that will pulse a 555 timer to drive an ignition coil. If you are good at soldering and circuitry, this way probably makes a nicer spark.
I answer a lot of tesla coil and Nikola Tesla questions, so you might want to browse thru my profile for more info.