Say hello to 'Project Lee' your console destroyer.
£1294.91
PCPartPicker part list / Price breakdown by merchant
CPU: AMD - Ryzen 5 1600X 3.6GHz 6-Core Processor (£191.94 @ Aria PC) CPU Cooler: Noctua - NH-L9i 33.8 CFM CPU Cooler (£34.46 @ Ebuyer) Motherboard: ASRock - X370 Gaming-ITX/ac Mini ITX AM4 Motherboard (£147.82 @ More Computers) Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory (£172.37 @ Amazon UK) Storage: Western Digital - Blue 250GB 2.5" Solid State Drive (£82.99 @ Box Limited) Storage: Seagate - Barracuda 2TB 3.5" 7200RPM Internal Hard Drive (£53.50 @ Aria PC) Video Card: Zotac - GeForce GTX 1070 8GB Mini Video Card (£349.99 @ Overclockers.co.uk) Case: Silverstone - RVZ02B HTPC Case (£79.99 @ Overclockers.co.uk) Power Supply: Silverstone - 600W 80+ Gold Certified Fully-Modular SFX Power Supply (£99.11 @ Alza) Operating System: Microsoft - Windows 10 Home OEM 64-bit (£82.74 @ Aria PC) Total: £1294.91 Prices include shipping, taxes, and discounts when available Generated by PCPartPicker 2017-10-28 17:55 BST+0100
That case is absolutely amazing, should mimic the form factor, and looks of a console with great airflow. (slightly bigger than a PS3)
The motherboard has a wireless antenna in it which should allow you to use wifi and not hook it up to a modem. However, as always a wired connection is better than wifi. (console, pc etc)
Fully modular power supply adds less clutter, leaving out wires you won't need.
Things to shop for:
Keyboard & Mouse
Wireless controller, OR, buy a micro sd cable that's long enough to wire your PS4 controller into it. (I have a wired PS4 controller for my PC)