Depends on the resolution of your monitor.
For your computer, you can't run much more than a Radeon HD7750 or a Radeon HD6670 depending on price.
Here's what I would do. I would purchase a 6670 or 7750 for your current computer. If it works out great, you have a cheap solution to your problem.
If not, save up more for a proper gaming computer and while you save up, use the computer with the GPU you just bought.
Then when it comes to sell your old computer, it can be worth more and you can sell it as an upgraded gaming computer.