I would...
Actually go Intel i3-4370 with an appropriate LGA1150 board. Pricewise - about the same, and it's dual core with hyperthreading with a pretty high base clock. It'll be pretty fast.
That being said, if it's strictly between FX6300 and A10-7850k, you're already getting a dGPU, so you can go right to the FX6300 as the hex-core will offer more performance than the 7850 overall.
Unless you have a reason for sticking with AMD (as in you have a mobo lying around) - the i3 is the better performing part for only a small amount of extra cash overall. Plus you have some upgrade options to even better performance with the i5 or i7 CPUs. Meanwhile, the FX6300 is only bested by the FX8xxx/9xxx series and the A10-7850 is the top of the line quad core for AMD which is still slower overall than the dual core i3. The FX6300 and A10 are both unlocked, but that just leads down another rabbit hole with overclocking and more parts.
Just as a comparison:
i3-4670/Gigabyte board/8gb RAM: $227.00
FX-6350/Gigabyte board/8gb RAM: $203.00
A10-7870k/Gigabyte board/8gb RAM: 213.00
For this, I picked the best processor for the series, and the least expensive Gigabyte board for each socket (I've always found Gigabyte boards to be very reliable and solid, plus it keeps it fair), and all have 8gb of Crucial Ballistic Sport 2x4GB DDR3-1600mhz RAM.
Overall, as mentioned, the i3 setup is only $20.00 higher than the lowest AMD setup. Gaming tests at 1080p between the i3 and the A10 show the i3 beating it in everything with a GTX770 dGPU, so I would rule out the A10. So - the FX-6350 would outperform the A10 I'm pretty sure, leaving it somewhat of a toss up. The other thing to consider though is the AMD setup is significantly hotter (90+ watts) while the Intel i3 is only a 50w processor.
So - all things given, the i3 would be the best overall performer, but a little more expensive, but a cooler chip. it would be my first choice.
Second choice would be the FX6300/6350. Perfectly good CPU, and a little cheaper than either the A10 or i3 setup.
Last choice would be the A10. Higher price than the FX, cheaper than the i3. Same thermals as the FX6300. Lowest performance of the three.