First off, this isn't exactly the best researched, maximum bang for your buck build... but, at the very least, it'll work well as a starting point for your build.
A few notes:
You said you wanted AMD, so that's what I went with. I ignored your comment about wanting to use an A10 APU because there's really not a lot of point in going for one of the APUs if you're going to be using a decently capable discrete graphics card. The APUs are a great choice if you're not planning on having a discrete graphics solution, however.
The aftermarket heatsink is optional, but it'll keep things cooler than the stock unit and should help to quiet things down. Also, pick whatever computer case you want... the one I selected is just a placeholder/suggestion.
Do not skimp on your power supply.
I cannot emphasize this enough. Too many people buy cheap, low quality PSUs and then wonder why they have all sorts of stability problems with their system.
If you can stretch your budget, I'd suggest using an SSD as a boot drive (my recommendation would either a Samsung 830 or Crucial M4) and the platter HDD as a storage drive... SSDs really make a MASSIVE difference in overall system performance but are still somewhat expensive.
Subtotal for the above is $742.91