There are only a handful of CPUs that make sense:
i3 6100 - $120
i5 6500 - $200
i5 6600K - $220 + cooler
i7 6700 - $300
i7 6700K - $320 + cooler
The i3 will perform the same as the i5's and i7's in 90% of games. The i5 will perform the same as an i7 in 99% of games. The i7 is useful if you intend to do other things while gaming, such as leaving video capture running in the background. Most people consider an i3 to be the sweet spot for a budget PC that still performs very well, an i5 to be the sweet spot for more serious gaming, and an i7 the sweet spot for a power user who often leaves lots of computationally expensive stuff open all the time.
Any of these CPUs will work in any socket 1151 motherboard. H110 chipset boards are limited to 2 RAM slots and 1 PCIe 16x slot. B150 and H170 boards are a little more expensive and get more slots and connectors, but are otherwise the same. Z170 boards, when paired with a "K" CPU, allow overclocking. You can get a Z170 board for about $80, and an H110 board for about $50.
Overclocking setups tend to offer worse value but higher absolute performance. You have to spend $30 more on a motherboard, and the CPUs don't come with their own cooler, so there's another $30. They're also $20 more expensive than their non-K counterparts, adding a total of about $80 to your build for the "privilege" of overclocking. You can expect to get 10-15% extra performance from overclocking to your chip's limits.