You definitely want to go with an NVidia GPU 650 Ti Boost or better. Since free tool from Nvidia called shadowplay will actually let you do the video encoding at 1080p 60fps in real time. You are looking at taking approximately 10% performance hit from it for your frame rates but that is not a huge problem.
This will probably work fine. Windows 7 included. If you can find an extra $50 somewhere, i would recommend upgrading the GPU to GTX760.
PCPartPicker part list:
http://pcpartpicker.com/p/2xVsY
Price breakdown by merchant: http://pcpartpicker.com/p/2xVsY/by_merchant/
Benchmarks: http://pcpartpicker.com/p/2xVsY/benchmarks/
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($109.99 @ Newegg)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: Asus M5A97 LE R2.0 ATX AM3+ Motherboard ($76.97 @ OutletPC)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1866 Memory ($68.00 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.98 @ OutletPC)
Video Card: MSI GeForce GTX 660 2GB Video Card ($179.99 @ Microcenter)
Case: Corsair 200R ATX Mid Tower Case ($49.99 @ NCIX US)
Power Supply: XFX 550W 80+ Bronze Certified ATX Power Supply ($49.99 @ NCIX US)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($19.98 @ OutletPC)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($88.98 @ OutletPC)
Total: $733.85
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-01-08 17:38 EST-0500)