$750-$850 Gaming PC Build

Hi there.

The Define Mini C is actually made by Fractal Design (not Corsair as you mentioned). Anyway, here is my suggested build (good quality, compatibility, and performance) based on your requirements and budget:

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-7500 3.4GHz Quad-Core Processor ($188.89 @ OutletPC)
CPU Cooler: CRYORIG M9i 48.4 CFM CPU Cooler ($19.88 @ OutletPC)
Motherboard: MSI B250M PRO-VDH Micro ATX LGA1151 Motherboard ($69.99 @ SuperBiiz)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($101.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($48.88 @ OutletPC)
Video Card: Gigabyte GeForce GTX 1060 6GB 6GB WINDFORCE OC 6G Video Card ($249.99 @ SuperBiiz)
Case: Fractal Design Define Mini C with Window MicroATX Mid Tower Case ($79.99 @ SuperBiiz)
Power Supply: EVGA SuperNOVA G3 550W 80+ Gold Certified Fully-Modular ATX Power Supply ($79.99 @ SuperBiiz)
Total: $839.60
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-04-25 06:04 EDT-0400

CPU = Got the 4-core/4-thread Kaby Lake (7th-gen) Intel i5-7500 for best price/performance, lower TDP (heat/power consumption), and fully capable of handling powerful GPUs for gaming without bottlenecks.

CPU Cooler = Replaced the included Intel stock cooler (that comes with the CPU box) to an affordable but significantly better thermal dissipation and noise reduction CPU cooler.

MB = To pair with the non-OCable i5-7500, a B250-chipset motherboard was selected that supports Kaby Lake CPUs out-of-the-box without any need for BIOS updates. This motherboard is mATX in size, has 4 DIMM (RAM) slots supporting up to 64GB and 2400MHz speeds, and a PCIe x16 slot (for your GPU) equipped with metal support frame bracket (for stability). It includes all the necessary/basic features that you would need for ports/connectivity (M.2 drive, SATA ports, USB2.0 and USB3.0 headers, etc.)

RAM = Got a 16GB (2 x 8GB) dual-channel 2400MHz memory sticks which is supported by the MB. The 16GB is very ideal in this day and age (although you can still get by with an 8GB or 2 x 4GB sticks - but that is now considered the minimum as far as gaming is concerned - in case you want to save more money).

Storage = Got a 1TB WD Blue, a good price/performance Hard Disk Drive (HDD) for all your storage needs. It's an option to also get a 250GB Solid State Drive (SSD) to be used specifically for your system drive (OS) for faster boot times but the SSD is just gravy and won't see any significant increase in gaming performance. You can add an SSD at a much later time if you have the budget for it. Otherwise, the HDD is enough for gaming.

GPU = Got the 6GB version of the GTX 1060 with backplate (for stability, protection, and better looks). The GPU fits well into the case you want.

Case = As preferred, the Fractal Design (not Corsair) Define Mini C was chosen -- the one with the window version to show off your build (black/gray theme).

PSU = Got you a very high-quality power supply unit that has more than enough amps/watts (45.8A at +12V rail) for the rig you need as well as future upgrades (to a more powerful i7 or GTX 1080) should you choose to do so without changing the PSU. The EVGA SuperNOVA G3 (550W) is made by Super Flower, a well-renowned manufacturer based on their Super Leadex II Gold Platform. The PSU is fully-modular (you only plug in the cables you need), has ample PCIe connectors for your GPU (as well as for the future GPU upgrades), has a 50C rated oper. temp, has a Fluid Dynamic Bearing fan, and has 7-year warranty.

Hope this suggestion helps! Good luck in your build!