With the a8-3870k, by running a standalone video card, it should take load off the CPU thus increasing the performance, correct?
Well, if you run a Radeon 7750 or better, you won't be using the integrated graphics on the cpu at all, which is why the Athlon 651K is a decent choice if you can find it cheaper than the 3870K, though the integrated video is nice as a backup. The performance will be better with a stand alone card because the card itself is better, and the dual graphics mode with a 6670 doesn't always increase performance, depending on game coding and drivers.
As far as the 7850 goes, it sort of stands alone in its price-point. It's a good card, especially if you can score a rebate deal for ~$150. Don't pay more than 170. As long as your psu is rated 4X0w or more and is 80+ or 80+ Bronze certified, the 7850 will be fine.
Here are a couple psu deals:
$40 - a $20 rebate = $20: http://www.newegg.com/Product/Product.aspx?Item=N82E16817139026
$30 after rebate: http://www.newegg.com/Product/Product.aspx?Item=N82E16817371030
The 6870 (as well as a Geforce 560 (non-se, non-ti)), either ties or edges out a Geforce 650ti due to the greater memory bandwidth, but it's older tech now and isn't as strong in DX11 titles. The 650ti also uses much less power. The 7850 will outperform all of those, though.
One other thing to note is that you probably won't be able to overclock with that motherboard. I imagine the bios is quite limited. Although it's possible that the AMD overdrive software will do it. I haven't actually tried it.
Also for convenience, here is a sort of FM1 cpus. The 651K is $88; the 3870K is $98. http://pcpartpicker.com/parts/cpu/#xcx=0&k=20&sort=a7
Note that, for those prices...you're *almost* to the point that you might want to do a new board and cpu instead of upgrade. Wait, here ya go, 3870K for $75 on Ebay, new in box: http://www.ebay.com/itm/New-Sealed-in-Box-AMD-A8-3870K-Socket-FM1-Black-Edition-3-0Ghz-Free-Shipping-/281056804499?pt=CPUs&hash=item41704a7e93