This isn't a great build but it'll do what you asked, and is the best performance you're gonna get for that budget. Expect it to perform noticeably better than any console. 1080p 60fps at medium settings in just about anything (except for maybe Crysis 3) and it should pound Dota and CS:GO into the dust.
PCPartPicker part list:
http://ca.pcpartpicker.com/p/J9gXcf
Price breakdown by merchant: http://ca.pcpartpicker.com/p/J9gXcf/by_merchant/
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($128.38 @ DirectCanada)
Motherboard: Asus M5A78L-M/USB3 Micro ATX AM3+ Motherboard ($62.50 @ Vuugo)
Memory: Kingston 4GB (1 x 4GB) DDR3-1600 Memory ($28.54 @ DirectCanada)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($59.75 @ Vuugo)
Video Card: MSI Radeon R7 370 2GB Video Card ($194.25 @ Vuugo)
Case: Rosewill RANGER-M MicroATX Mini Tower Case ($64.98 @ Newegg Canada)
Power Supply: Corsair CX 430W 80+ Bronze Certified Semi-Modular ATX Power Supply ($45.98 @ NCIX)
Monitor: Acer UM.WV6AA.B01 60Hz 21.5" Monitor ($125.50 @ Vuugo)
Total: $709.88
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-10-03 10:52 EDT-0400