1,600 USD? For Pure gaming? Heh, you're gonna get a beast of an upgrade.
[PCPartPicker part list](
http://pcpartpicker.com/list/gCj89W) / [Price breakdown by merchant](
http://pcpartpicker.com/list/gCj89W/by_merchant/)
Type|Item|Price
:----|:----|:----
**CPU** | [Intel Core i7-6700K 4.0GHz Quad-Core Processor](
http://pcpartpicker.com/product/tdmx...bx80662i76700k) | $324.99 @ SuperBiiz
**CPU Cooler** | [Corsair H100i 77.0 CFM Liquid CPU Cooler](
http://pcpartpicker.com/product/V2Jw...u-cooler-h100i) | $139.89 @ OutletPC
**Motherboard** | [MSI Z170A SLI PLUS ATX LGA1151 Motherboard](
http://pcpartpicker.com/product/yLPz...d-z170asliplus) | $137.84 @ OutletPC
**Memory** | [Mushkin Blackline 16GB (2 x 8GB) DDR4-2400 Memory](
http://pcpartpicker.com/product/xWZ2...memory-997199f) | $61.99 @ Newegg
**Video Card** | [EVGA GeForce GTX 1080 8GB FTW Gaming ACX 3.0 Video Card](
http://pcpartpicker.com/product/bNcM...card-08gp46286) | $679.99 @ B&H
| *Prices include shipping, taxes, rebates, and discounts* |
| **Total** | **$1344.70**
| Generated by [PCPartPicker](
http://pcpartpicker.com) 2016-09-20 19:18 EDT-0400 |
That's what I'd buy. BUT.
What Zed said is partially true. The i7-6700k IS overkill for 1080p gaming. The 980Ti is also perfectly fine for 1080p.
But, with the left-over money, you could roll into a 3x1080 Surround setup with a GTX 1080. That would look beautiful, and you could do it with Triple Asus VG248QE's to get 144hz 3x Surround 1080p. That would be buttery smooth.
If you stuck with an i5-6600k and said screw getting a Kabylake and new monitors, you could go all the way to a Titan X(Pascal) within your budget.
Kabylake isn't even going to be a huge upgrade over Skylake, so personally, i'd just go with the i5-6600k and a 1080. And upgrade to 3x Asus VG248QE's for Surround 1080p buttery smoothness.