Strictly gaming...? Definetly an i5 6600k right now.
I think an i5 build with an RX 480 is probably your best bet. You can probably get it under $800...not including Windows or a Display. Start by looking at the builds in the OT.
Something like this...?
PCPartPicker part list:
http://pcpartpicker.com/list/DyxJ7h
Price breakdown by merchant:
http://pcpartpicker.com/list/DyxJ7h/by_merchant/
CPU: Intel Core i5-6600K 3.5GHz Quad-Core Processor ($233.99 @ SuperBiiz)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($24.89 @ OutletPC)
Motherboard: Gigabyte GA-Z170-HD3P ATX LGA1151 Motherboard ($102.99 @ SuperBiiz)
Memory: Corsair Vengeance LPX 16GB (2 x 8GB) DDR4-2400 Memory ($57.99 @ Amazon)
Storage: Sandisk SSD PLUS 480GB 2.5" Solid State Drive ($94.99 @ Amazon)
Video Card: MSI Radeon R9 380 4GB Video Card ($199.99 @ Micro Center)
Case: Corsair 200R ATX Mid Tower Case ($44.99 @ Newegg)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($42.99 @ SuperBiiz)
Total: $802.82
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-02 17:52 EDT-0400
I put an R9 280 in there as a stand in for the 480...similar price.