Ok - here is the best build that you can get, with a monitor, for less than $800.
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | Intel Core i3-4330 3.5GHz Dual-Core Processor | $135.98 @ Newegg |
| Motherboard | Gigabyte GA-B85M-DS3H Micro ATX LGA1150 Motherboard | $55.98 @ Newegg |
| Memory | Team Zeus Blue 8GB (2 x 4GB) DDR3-1600 Memory | $59.99 @ Newegg |
| Storage | A-Data Premier Pro SP600 128GB 2.5" Solid State Drive | $59.99 @ Amazon |
| Storage | Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive | $49.88 @ OutletPC |
| Video Card | Sapphire Radeon R9 270X 2GB Vapor-X Video Card | $168.99 @ SuperBiiz |
| Case | Fractal Design Core 1500 MicroATX Mini Tower Case | $39.99 @ NCIX US |
| Power Supply | Cooler Master 550W 80+ Bronze Certified Semi-Modular ATX Power Supply | $79.99 @ Mac Mall |
| Optical Drive | Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer | $18.75 @ OutletPC |
| Monitor | Acer G227HQLbi 60Hz 21.5" Monitor | $119.98 @ OutletPC |
| Total | ||
| Prices include shipping, taxes, and discounts when available | $780.52 | |
| Generated by PCPartPicker 2014-12-30 10:17 EST-0500 |
If you are primarily gaming, it is better to have fewer but stronger CPU cores/threads rather than more, weaker cores/threads thus, the i3 is a better option, for you, than the 6300.
I've gone for a nicer case because that Azza would be pretty nasty tbh.
I've also got an SSD in there - it's definitely worth it, just get one, you won't regret it.
I've also gone for a nicer monitor and a semi modular PSU.