If you don't mind building your own computer, this should work out well for you
CASE: You can choose any you personally like but for a budget computer, this would be a good one since it's cheap and it works and has space in the back for you to tuck and hide your cables
NZXT Source 210 at $29.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16811146076
MOTHERBOARD: MSI Z77A-G41 at $94.99 (not including the $10 mail in rebate yet):
http://www.newegg.com/Product/Product.aspx?Item=N82E16813130653
PROCESSOR: i3-3220 at $129.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116775
RAM: G.Skill Ripjaws X Series DDR3-1600 8GB (2x4GB) at $49.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231428
POWER SUPPLY: Corsair CX430 at $46.99 (not including the $20 mail in rebate yet):
http://www.newegg.com/Product/Product.aspx?Item=N82E16817139026
GRAPHICS CARD: any amd radeon 7770 should be fine for your purpose. I personally chose the XFX Double D 7770 (model FX-777A-ZDF4) at $139.99 (not including the $30 mail in rebate yet):
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150597
STORAGE: I assume you can reuse the current hard drive(s) you have. If you need more storage, you can get this Western Digital Green 1TB (model WD10EZRX) at $79.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16822236070
DVD/CD DRIVE: I assume you can reuse the current dvd/cd drive you have. If for whatever reason, your current one is broken and you need a new one, you can get this ASUS DVD/CD drive at $19.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16827135204
I also assume you are going to be using the same monitor, keyboard, and mouse
Adding everything up except for the storage and dvd/cd drive section yields a total cost of $491.94. The mail in rebates add up to $60. Tax and shipping have not been accounted for.
Adding everything up including the storage and dvd/cd drive section yields $591.92. The mail in rebates still add up to $60. Tax and shipping have not been accounted for.
OH! And if you need an operating system and don't have an installation disc already or do not plan to get it for free from other "sources"...
Windows 7 Home Premium 64 bit at $99.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16832116986
Windows 8 64 bit at $99.99:
http://www.newegg.com/Product/Product.aspx?Item=N82E16832416550
I highly recommend windows 7 (personal preference though).
Anyways, adding the OS will just bring up the cost by $100, $591.93 and $691.91 respectively.
It really isn't that hard to build a computer. Sure, it won't look as great as someone with experience already but it's actually pretty fun. You'll save some money too by building it yourself. If you really must buy a pre-built computer, I will try and see if I can find a decent one at the budget you have
For the budget you have, it's more beneficial to just build your own computer.
edit: there are PLENTY of youtube videos out there teaching you how to build/put together a computer. Here's from from newegg. (yes, a bit lengthy but it goes into quite the detail)
part 1 (choosing your components):
http://www.youtube.com/watch?v=lPIXAtNGGCw
part 2 (the build):
http://www.youtube.com/watch?v=d_56kyib-Ls
part 3: (installing windows and finishing touches):
http://www.youtube.com/watch?v=RxaVBsXEiok
edit2: I personally cannot fathom how expensive these desktops are (for the hardware given) so I cannot make any good pre-built desktop recommendations to you. Hopefully another individual can help you with that