You can go cheaper than that on Skylake.
PCPartPicker part list:
http://pcpartpicker.com/p/ZFcY4D
Price breakdown by merchant: http://pcpartpicker.com/p/ZFcY4D/by_merchant/
CPU: Intel Core i5-6600K 3.5GHz Quad-Core Processor ($249.99 @ Newegg)
Motherboard: ASRock Z170M Pro4S Micro ATX LGA1151 Motherboard ($99.99 @ SuperBiiz)
Memory: G.Skill Aegis 8GB (2 x 4GB) DDR4-2400 Memory ($39.99 @ Newegg)
Total: $389.97
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-02-10 22:08 EST-0500
You don't need an aftermarket cooler for a Skylake build, the one Intel includes will be fine. You won't be able to overclock with it, but you can get the basic build then buy a cooler later when you want to overclock. The Core i5 is already a major jump up in performance for you compared to the FX 6300, and you really don't need to go any higher. The i7-6700K does have better performance and will last longer because Hyper-Threading extends the performance significantly, but it all depends on your budget. This build isn't as good as the one recommended by DonkeyOatie in terms of performance, but it still has a decent quality motherboard, a high performance CPU, and an alright RAM kit so if you don't have too much cash, this build is probably best.
If you do have the extra cash to toss into it, DonkeyOatie's skylake build suggestion is better, just more expensive.