You should look into a mini ITX build. They can be tough for beginners because they're small cases, but there are plenty of tiny cases that will allow you to fit a high-end graphics card. Your build budget would look something like this:
Case: $75 - Lian Li and Silverstone have some great mini ITX cases on newegg.com, the site I use to buy components.
Storage: $70 for a 1TB HDD
GPU: $300 for a GTX 660 or similar
MoBo: $90 Search for Mini ITX Motherboard. ASRock makes a great H77 model
PSU: $70 You need a 500W modular PSU made by Seasonic, Antec, or PC Power and Cooling
Memory: $50 get the cheapest two 4GB 240pin DDR3 sticks you can find. Usually Crucial has the lowest price offering.
CPU: $170 Grap the cheapest Ivy Bridge i5 you can find. i3 will be sufficient if you're not worried about Directx 11 games.
Disc Drive: $20
OS: $80
Total is $900ish and that will run anything.