First, would you be willing to build your own PC?
Anyway, I'd recommend any tower that comes with an Intel i3/i5 or AMD FX Quad/Six Core CPU and at least 6gb RAM. I'd go with HP, Asus, or Acer/Gateway.
You are better off not worrying about what GPU a tower brings (so don't look for one that already brings one) and buy your own. An Nvidia GTX560 goes for about ~$190, so this plus a ~$500 tower would be a great pair. Finally get yourself a 500w power supply. The total should be around your budget.
You can also wait for the GTX660 due out next month, but it will be pricey. If want to go AMD for your GPU, then I advise the 6870.
Edit: Heres an example
http://www.newegg.com/Product/Product.aspx?Item=N82E16883113203
e=mc^2
Gaming on: PS4 Pro, Switch, SNES Mini, Wii U, PC (i5-7400, GTX 1060)
