The budget for the monitor matters since you'll want the PC and monitor to balance well together in terms of spec.
And for PVP shooters with this kind of budget you'll ideally want a 1080p 144hz screen which might often be ~£180-200. And a PC good enough to drive it.
You may well end up playing on generally low settings to try to maximise framerates for a competitive advantage.
In terms of the build:
PCPartPicker Part List
CPU: AMD Ryzen 5 3600 3.6 GHz 6-Core Processor
(£158.97 @ Aria PC)
Motherboard: ASRock Fatal1ty B450 GAMING K4 ATX AM4 Motherboard
(£83.99 @ CCL Computers)
Memory: Team T-FORCE VULCAN Z 16 GB (2 x 8 GB) DDR4-3200 CL16 Memory
(£57.39 @ Amazon UK)
Storage: Crucial BX500 480 GB 2.5" Solid State Drive
(£49.37 @ CCL Computers)
Video Card: PNY GeForce GTX 1650 SUPER 4 GB Video Card
(£156.99 @ Amazon UK)
Case: Fractal Design Focus G ATX Mid Tower Case
(£49.99 @ Currys PC World)
Power Supply: be quiet! System Power 9 600 W 80+ Bronze Certified ATX Power Supply
(£54.26 @ CCL Computers)
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2020-08-10 21:40 BST+0100
PSU pricing and availability is messed up by the pandemic, so the 600W Bronze unit there is actually one of the better buys right now it seems.
A 1660 Super would be ideal, especially if you intend to play more demanding games at higher settings too. The 1660 non-Super is not worth considering as it's only £20 cheaper than the Super. The 1660 Super does take the price of the system minus monitor to ~£660, but it could be worth looking at extending the budget for that.
If buying on Amazon, only buy items sold buy Amazon themselves, not marketplace sellers, as it could potentially cause warranty issues.
Monitor options include: