OK,
If you are doing more than just gaming (video editing, photo editing, etc.) this is what I would suggest:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 7 1700 3.0GHz 8-Core Processor (£269.94 @ Aria PC)
CPU Cooler: CRYORIG - H7 49.0 CFM CPU Cooler (£34.99 @ Overclockers.co.uk)
Motherboard: MSI - B350 PC MATE ATX AM4 Motherboard (£80.86 @ CCL Computers)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3000 Memory (£116.77 @ Amazon UK)
Total: £502.56
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-25 05:47 BST+0100
Rationale:
CPU: The R7 1700 has eight cores and a good IPC.
CPU cooler: My normal suggestion at this price. A good cooler which you should be able to get to decent clocks with.
MOBO: Nice and cheap while being full ATX and all the features you need!
RAM: 2x8gb at 3000Mhz for a decent price.
If you are just doing gaming and web-browsing etc., I would recommend this:
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD - Ryzen 5 1600 3.2GHz 6-Core Processor (£195.42 @ Ebuyer)
CPU Cooler: Noctua - NH-D15 SE-AM4 140.2 CFM CPU Cooler (£78.95 @ Amazon UK)
Motherboard: MSI - B350 PC MATE ATX AM4 Motherboard (£80.86 @ CCL Computers)
Memory: G.Skill - Ripjaws V Series 16GB (2 x 8GB) DDR4-3200 Memory (£123.85 @ Amazon UK)
Total: £479.08
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-25 05:53 BST+0100
Rationale:
CPU: 6 fast cores and a decent price
CPU cooler: as good as you can get with air, provided you have a large enough case (you may want to check that!), you should be able to get your CPU to way over 4Ghz
MOBO: same
RAM: Slightly faster and costs slightly more than above.
If you are gaming at over 120hz, then you will need an i7:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i7-7700K 4.2GHz Quad-Core Processor (£298.99 @ Aria PC)
CPU Cooler: CRYORIG - H7 49.0 CFM CPU Cooler (£34.99 @ Overclockers.co.uk)
Motherboard: MSI - Z270 PC MATE ATX LGA1151 Motherboard (£102.73 @ Amazon UK)
Memory: G.Skill - Ripjaws 4 series 8GB (2 x 4GB) DDR4-2400 Memory (£59.30 @ Amazon UK)
Total: £496.01
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2017-06-25 06:01 BST+0100
Rationale:
CPU: the highest IPC on the market right now, you still can't beat an i7 for high refresh rate gaming.
CPU cooler: the only decent thing I could fit in the budget (the 7700k does not come with a cooler)
MOBO: the cheapest thing that can overclock on this platform
RAM: I couldn't fit any more than 8gb in the budget!
Hopefully that helps!