Yeah, it's possible. Despite appearances, PC power supplies are actually pretty high quality due to the fact that computer components are very sensitive to fluctuations in voltage. They also tend to have built in current protection.

First thing to check is the amperage on the 12v rail. It's usually specified on a sticker on the side. You'll need a PSU which supplies enough amperage to your charger.

ATX PSU's have a soft switch - they turn on when they detect a load. You'll need to wire up a resistor to force it to power up.

Finally, you need to solder some connections to the 12v lines.

Here is a good article with step by step instructions. Here are more details about the one I made.