Alright, this is the build I've come up with:
Corsair CX550M, 550W PSU
Intel Core i5-6400, Socket-LGA1151
ASUS B150M-A, Socket-1151
Crucial DDR4 2133MHz 8GB (2x4GB)
XFX Radeon RX 480 XXX OC
Seagate Barracuda® 1TB
There wasn't enough money to squeeze in an SSD after all, but I think it'll be fine.