You can easily get on AM4 with that budget and it leaves you with an upgrade path to cheap 6/8 core CPUs in the future, as well as Ryzen 3000 early next year.
I'd suggest just sticking with that otherwise you have to try and find an i7 4790K for a cheap price and hope it has a decent Z chipset board for overclocking, unless that was like $150 or less I'd just stick to AM4, it'll be fine for 60-120hz gaming depending on the game. I assume you only have a 1080p 60hz monitor at best.
PCPartPicker part list: https://uk.pcpartpicker.com/list/VXNvq4
Price breakdown by merchant: https://uk.pcpartpicker.com/list/VXNvq4/by_merchant/
CPU: AMD - Ryzen 3 2200G 3.5GHz Quad-Core Processor (£86.39 @ Aria PC)
Motherboard: ASRock - B450M PRO4 Micro ATX AM4 Motherboard (£71.99 @ Amazon UK)
Memory: Team - Vulcan 8GB (2 x 4GB) DDR4-3000 Memory (£68.99 @ CCL Computers)
Total: £227.37
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-10-24 20:37 BST+0100