Your machine is a slim model, with only a 220W PSU. It isn't a standard ATX PSU, so you won't find one much more powerful.
This will severely limit the graphics card you can install. A HD5570 is probably it, like this one: http://www.newegg.com/Product/Product.aspx?Item=N82E16814102874
which comes with a low-profile bracket.
The good news is that you'll spend only a fraction of your budget. The bad news is, it isn't going to be great for the latest games.
If you want to play games, a better choice might be to relocate the components of that PC into a regular mid-tower case (even a cheap one), put a 380W Antec Earthwatts PSU in it, then get a HD5750 graphics card. You'll spend around $200 for that option, mostly depending on the cost of the case. If you are entirely ok with spending the full $300, use a 500W Antec Earthwatts, and put in a HD5770.