Are you looking for a certain budget to hit at most, or just as low as you can go? Do you need a monitor? What resolution are you going to game at, typically? Do you need an Operating System to go with it?
As it is, I would personally suggest getting a non-Trinity system and just run off a more powerful GPU than to use integrated and a lower-tier GPU like a 6670. With what you have listed there, unless you're getting the 600w PSU for fairly cheap (as in, under $50), you could probably get away with a 430w PSU instead. I'd normally recommend 8GB of RAM, but in a budget build, 4GB should be decent. Make sure your mobo chipset of choice is either FM2 for the A10, or AM3+ if you switch to something like an FX-4300 or -6300. Some may also recommend switching brands unless you prefer MSI- most people seem to prefer ASRock, but Gigabyte is pretty good as well. Up to you, since similar boards to the one you chose are roughly the same price.
If you are interested in the FX route, here's an FX-4300 build.
http://pcpartpicker.com/p/T6Lg Going with an FX-6300 instead is just $20 more, and is comparatively priced with the AMD A10-5800k. In fact, from what I put in of your build into PCpartpicker, the FX-4300 build is actually a little cheaper than the A10 build (after rebates 'n such). I didn't expect that, especially since it has a more expensive graphic card and 8gb of RAM.
The monitor's resolution is going to be an issue. For higher resolution monitors, you may not even hit medium settings on current games. On lower resolutions (like 1600x900 or less, I believe), you should be okay at medium in most cases. I'm not an expert on Trinity systems by any means (my PC is an Intel build), but I think a single dedicated, higher priced GPU would do better than a crossfire setup between the APU and a GPU that's compatible.
Anyway, there ya go.