I've gone a lot under budget because you didn't really provide information about what your budget has to include, so you have a bit of flexibility going on here.
PCPartPicker part list /
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CPU: Intel Core i5-4670K 3.4GHz Quad-Core Processor ($209.99 @ Amazon)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: ASRock Z87E-ITX Mini ITX LGA1150 Motherboard ($139.99 @ Newegg)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1600 Memory ($68.99 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($54.98 @ OutletPC)
Video Card: MSI GeForce GTX 780 3GB Video Card ($469.99 @ Newegg)
Case: BitFenix Prodigy (Black) Mini ITX Tower Case ($59.99 @ NCIX US)
Power Supply: Antec High Current Gamer 620W 80+ Bronze Certified Semi-Modular ATX Power Supply ($59.99 @ NCIX US)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($15.98 @ OutletPC)
Operating System: Microsoft Windows 8.1 (OEM) (64-bit) ($89.98 @ OutletPC)
Total: $1199.86
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2014-04-20 02:19 EDT-0400)
CPU+Platform - An overclocking build isn't essential, and if it's something you aren't interested then you can comfortably downgrade the CPU+motherboard+cooler as applicable. Equally, if you want to push the overclock further, you can upgrade the board to say an Asus Z87I-Deluxe and Noctua NH-D14.
Memory+Storage - Both can be upgraded if needed. For an exclusively gaming build 8Gb of 1600Mhz memory is absolutely fine, but given the OC build you might want to invest in quicker memory. Similarly, an SSD can be very tempting and at your budget level it's possibly advisable. Something like a Samsung 840 EVO 120Gb is quick, reliable and fairly affordable.
Graphics - You can spend as much as you like here, but it's limited by your monitor display resolution as to how much makes sense to spend. For 1080P gaming, a GTX770 or lower is actually plenty, but going up to something like a GTX780 gives you more flexibility with larger resolutions and potentially more lifespan too.
PSU - Cheap, semi-modular, great quality (great OEM) and should fit fairly comfortably in a Prodigy. There's not many better options. It's more capacity than you need, but it's within the realm of sense and gives you some flexibility with which graphics card you chose.
Extras - If you do go with an OC build, I'd recommend replacing both of the stock fans on this case. The rear can be replaced by a quality 140mm, and the front can be upgraded to a 140/200/230mm, all of which are significantly better than the stock 120 as they blow straight across the motherboard.
Also if you need things like a monitor or peripherals then that'll easily swallow the rest of your budget.