Deliver a frame rate of 60 fps when running Overwatch at high settings? Probably yes.
PSU good enough for GTX 1060? Definitely yes.
But...Building a new PC, mostly to run games, using the FX-6300 in late 2016? Nope. Because, by doing so you will be closing all gates to an effortless and efficient future upgrade. As you'd know that the CPU in question requires the AM3+ socket and considering the new AMD Zen CPUs will need AM4 sockets (and probably any other new release in the near future will tread along that path) - you will need to swap your motherboard as well whenever you wish to upgrade your processor and if you choose the FX - 6300, you'd be wishing to upgrade much sooner than you may imagine.
If you are about to start working on your build, it would be much wiser to get a motherboard having LGA1151 socket and then deciding which CPU (latest gen i3/i5/i7) to use according to your requirement and budget, that way, you will have the option of just upgrading your CPU if required without much sweating (or spending) for the next few years. And if you are of the patient type, consider waiting for the AMD Zen CPU's release next year and see what it has got to offer.
Coming to the GPU, why would you wish to replace the R9 380X with the GTX 1060 within months? I have failed to understand the logic behind it. If I were in your place, instead of the R9 380X I would have installed either of the RX 470 or RX 480 or the GTX 1060 itself - all of them being relatively newer, better and within the same price range.
Also, more RAM has never done any harm to anybody, but to me, 16 GB RAM seems like an overkill at this moment as most games can be run effortlessly with 8 GB of it. You may wish to allocate the few precious bucks thus saved to the budget of your graphics card and consider getting additional RAM (thats why it might be a better idea to get a single stick of 8 GB than 2x4 GB) later when more demanding games that require more RAM are released eventually.
Hope it helped.