What CPU should I get with my budget?

[TactX21]

Hello everyone,
I am looking to upgrade my desktop PC after Christmas so I wanted to ask about what I should get. Currently I have AMD FX-4100 and my GPU is AMD Radeon HD 7870 2GB.
Would it be better to get an intel or AMD cpu, and which one should I get given my total budget will be about 300-400 pounds?
This budget also includes buying a new CPU cooler as I will need one, and probably a new motherboard because my current one most likely won't be compatible. Finally, I was thinking about also buying a new case if necessary, and I possibly could push up my budget if needed.
I appreciate any help and tips

 

[JQB45]

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor (£181.90 @ Amazon UK)
Motherboard: ASRock H170 Pro4 ATX LGA1151 Motherboard (£90.90 @ Scan.co.uk)
Memory: Kingston HyperX Fury Black 16GB (2 x 8GB) DDR4-2133 Memory (£67.99 @ Novatech)
Case: Corsair 200R ATX Mid Tower Case (£54.98 @ Novatech)
Total: £395.77
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-10-17 23:28 BST+0100

There currently is no AMD CPU that can beat a i5-6500 in single threaded applications or games that only use a few threads.
AMD Zen will be out early 2017 - you might hold off and see what the benchmarks look like.

If you don't like the case you may need to increase your budget and choose something you do like.
The i5-6500 comes with a CPU Cooler and Thermal Paste pre-installed.

 

[bwen1]

An i3 6100 will handle your graphics card fine, but not as future proof.