So, to summarize, here's the build as a one-stop-shop from Amazon (you'll save about $30 if you go through several merchants but may lose that in shipping). This includes appropriately less expensive DDR2133 to match the budget motherboard, gets you the good GPU you want, a quiet power supply, and includes the Xbox 360 wireless receiver.
You may want to post this draft build over at /r/sffpc as well for some advice.
PCPartPicker part list / Price breakdown by merchant
| Type | Item | Price |
|---|---|---|
| CPU | Intel Core i5-6500 3.2GHz Quad-Core Processor | $212.08 @ Amazon |
| Motherboard | ASRock H110M-ITX/ac Mini ITX LGA1151 Motherboard | $77.07 @ Amazon |
| Memory | Crucial 8GB (2 x 4GB) DDR4-2133 Memory | - |
| Storage | Sandisk Ultra II 480GB 2.5" Solid State Drive | $132.07 @ Amazon |
| Video Card | Asus GeForce GTX 1060 6GB Turbo Video Card | $279.77 @ Amazon |
| Case | Silverstone ML08B-H HTPC Case | $94.16 @ Amazon |
| Power Supply | Silverstone 600W 80+ Gold Certified Fully-Modular SFX Power Supply | $129.29 @ Amazon |
| Other | 4 Xbox Wireless Controllers | $136.00 |
| Other | Xbox 360 Windows Adapter | $18.00 |
| Prices include shipping, taxes, rebates, and discounts | ||
| Total | $1078.44 | |
| Generated by PCPartPicker 2016-09-12 05:08 EDT-0400 |