Is this a Good Gaming Build? (under $800)

Solution
If you are not going to overclock, then why get an unlocked CPU. Without overclocking, the build can be significantly cheaper.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ SuperBiiz)
Motherboard: ASRock H97M PRO4 Micro ATX LGA1150 Motherboard ($60.35 @ SuperBiiz)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1866 Memory ($44.99 @ Newegg)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive ($77.99 @ Amazon)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($39.10 @ SuperBiiz)
Video Card: EVGA GeForce GTX 960 2GB SuperSC ACX 2.0+ Video...
Better in lots of little ways, and less than your build, but still over $800.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4690K 3.5GHz Quad-Core Processor ($209.99 @ Newegg)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($19.99 @ Newegg)
Motherboard: Gigabyte GA-Z97X-Gaming 5 ATX LGA1150 Motherboard ($113.99 @ SuperBiiz)
Memory: G.Skill Trident X Series 8GB (2 x 4GB) DDR3-2400 Memory ($41.99 @ Newegg)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive ($77.99 @ Amazon)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($39.10 @ SuperBiiz)
Video Card: Asus GeForce GTX 960 4GB Video Card ($179.99 @ Newegg)
Case: Cooler Master N400 ATX Mid Tower Case ($47.60 @ SuperBiiz)
Power Supply: EVGA 750W 80+ Bronze Certified Semi-Modular ATX Power Supply ($39.99 @ NCIX US)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($14.99 @ Newegg)
Keyboard: Logitech G710 Wired Gaming Keyboard ($49.99 @ Best Buy)
Total: $835.61
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-11-28 18:49 EST-0500

Better motherboard for overclocking
Faster memory
Better, bigger and faster SSD
More memory on your GPU
Cheaper great case
Excellent, more powerful, but only Bronze PSU.
 


That would have been my recommendation too, if it did not increase the price. however with the Hyper 212 EVO being $20 at the moment, is it considerably more cost effective. When the prices go back up, the balance will shift.

 


Is that motherboard necessary if i'm not looking to overclock?
Was the other PSU enough and viable for my computer?
 
If you are not going to overclock, then why get an unlocked CPU. Without overclocking, the build can be significantly cheaper.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($159.99 @ SuperBiiz)
Motherboard: ASRock H97M PRO4 Micro ATX LGA1150 Motherboard ($60.35 @ SuperBiiz)
Memory: G.Skill Ares Series 8GB (2 x 4GB) DDR3-1866 Memory ($44.99 @ Newegg)
Storage: Samsung 850 EVO-Series 250GB 2.5" Solid State Drive ($77.99 @ Amazon)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($39.10 @ SuperBiiz)
Video Card: EVGA GeForce GTX 960 2GB SuperSC ACX 2.0+ Video Card ($179.99 @ SuperBiiz)
Case: Thermaltake Overseer RX-I ATX Full Tower Case ($67.99 @ Newegg)
Power Supply: XFX TS 550W 80+ Bronze Certified ATX Power Supply ($52.98 @ Newegg)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($14.99 @ Newegg)
Keyboard: Logitech G710 Wired Gaming Keyboard ($49.99 @ Best Buy)
Total: $748.36
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-11-29 11:48 EST-0500

The other PSU was more efficient, but lower performance. 650 W is more than enough. The PSU I suggested is more powerful (which you do not need, and less efficient, but higher performance.)

And I'd still prefer my case, but cases are a personal choice and now we're below your target it does not matter.
 
Solution
All the ideas above are good workable builds.
To add my suggestion, you can do a conservative overclock of a 4690k on a Gigabyte H81 HD3 motherboard..
And the stock cooler is actually enough to cool an i5
Both these options will save you money
 


I wouldn't count an MIR as an actual discount. Plus the CRYORIG is on sale right now for 4 bucks less.
Even so, it's only an $11 difference and IMO it's worth it.

But it looks like he's not actually planning to OC so I guess it doesn't really matter.
 
MIR is less over-all end-use cost. I use them all the time with my school builds, often getting $50 to $100 back on each system. It does require paperwork and it does take a while, but I have always gotten them without problems. (because I follow the 'rules' carefully and document everything)