AMD's A10-7890K is where I suggest you turn to for a processor. Running at 4.1GHz and yet having a modest sub-$170 price tag, it makes an excellent choice for a gaming system no matter what your budget.
This chip also comes with the AMD Wraith cooler, which, for a stock cooler, isn't bad at all.
Here we're having to rely on the built-in Radeon R7 GPU, but this is still more than enough to give you 30 frames per second gaming at 1080HD resolutions in many of the popular games.
Since this is a budget system there's no need to overspend on a motherboard. However, with that said, we still want a decent board that offers stability and plenty of room for tweaking. Here I'd go for the Gigabyte GA-F2A78M-HD2, because it offers pretty much everything you'd expect from a high-end board - 4K support, USB 3.0, power failure protection - for around $60 if you shop around.
What a budget gaming system needs is a lot of storage at a decent price, so I'm going for my favorite - the 1TB Western Digital Blue WD10EZEX, which comes in at a shade over $50.
As much as I feel that I can save money by going for cheap RAM, when I'm building performance or gaming systems I always feel like this is a ghost that's going to come back to haunt me at a later date.
There's a lot of different RAM to choose from, but since I've been having good success with G.SKILL RAM in high-end systems, I'm going to stick with it here. 8GB of G.SKILL Ripjaws X Series DDR3 2133 comes in at around $40, so it's ideal for this system.
You can pick up something like Zalman T5for $30. It's not much to look at, but it does the job. At the end of the day it's what's in the box that counts!
You can't get away from paying the Windows tax!
An OEM copy of Windows 10 Home costs $100, while Windows 10 Professional will set you back $140.
Since the APU we're fitting features a built-in Radeon R9 GPU you can do without a processor. However, if you want more gaming performance from this build I suggest you add an XFX Radeon R7 360 R7-360P-2255 2GB, which will add $115 to the bill, and considerably boost the graphics performance of the build.
Another way to boost the overall performance of this build would be to add an SSD, something such as the Mushkin Enhanced Reactor 1TB which you can pick up for around $250. This offers the following read/write speeds: