If the rendering is just recreational and not really of critical importance, you could just focus the PC on gaming and the rendering will just take a few extra seconds here and there. Not the end of the world if it is only a few per week.
I'd go for an i5 and R7 370, here's a benchmark which shows that the R9 270 (which is slightly less powerful than the 370) has no problems playing Battlefield 4 at high settings and 1080P.
http://www.tomshardware.com/reviews/radeon-r9-270-review-benchmarks,3669-4.html
There's a very low cost build that will quite happily game at 1080P and will be good for your rendering:
PCPartPicker part list:
http://uk.pcpartpicker.com/p/PMmLTW
Price breakdown by merchant: http://uk.pcpartpicker.com/p/PMmLTW/by_merchant/
CPU: Intel Core i5-4440 3.1GHz Quad-Core Processor (£138.52 @ Ebuyer)
Motherboard: MSI B85M-P33 V2 Micro ATX LGA1150 Motherboard (£40.59 @ Amazon UK)
Memory: Kingston HyperX Fury White 8GB (2 x 4GB) DDR3-1866 Memory (£29.99 @ Ebuyer)
Storage: Hitachi Ultrastar 750GB 3.5" 7200RPM Internal Hard Drive (£25.99 @ Aria PC)
Video Card: PowerColor Radeon R7 370 2GB PCS+ Video Card (£116.99 @ Ebuyer)
Case: Zalman ZM-T4 MicroATX Mini Tower Case (£14.99 @ Novatech)
Power Supply: Antec EarthWatts Green 380W 80+ Bronze Certified ATX Power Supply (£31.84 @ Amazon UK)
Total: £398.91
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-08-10 11:55 BST+0100
Couple of bargains there on the RAM and case in particular, leaves plenty of cash for a nice monitor. It should be stated though that you need to make sure that motherboard has an updated BIOS before buying it otherwise it may not be compatible with that CPU.
This seems to be a well reviewed 24'' monitor:
http://uk.pcpartpicker.com/part/benq-monitor-gl2460hm
Still some cash left over even after that so you could pocket it, buy a second monitor or upgrade the GPU/case to something more preferable. The GTX 960 is a very good card that will run well on that PSU.