A very good question.
I think the skylake I5 non k chips are priced badly.
I3-6100@3.7 $125 is an excellent mid range gamer.
I5-6600K $230 or so when overclocked(4.5 is assured) is as good as it gets for gaming today.
The intermediates like the i5-6500 @3.2 are not so competitive in games unless all 4 threads can be fully utilized. That is rare.
Here is a review of the i3-6100 where it competes favorably in the tested games with the more expensive i5-6600.
You have some options:
1. If you are on a strict budget, i3-6100 may well do the job for you.
Try to use a z170 motherboard to preserve your option to a future i5-6600K upgrade.
Or, use a cheaper H110 based motherboard and plan a upgrade to a i7-6700K which runs at 4.0 and needs no overclocking.
2. Z170 motherboard and i5-6600k. Plus, you will need a cooler since the k does not come with one.
3. If rumors are true, we are not too far from kaby lake. Perhaps January.
Of particular interest is the I3-7350K which will be $177.
runs 4.0 out of the box and allows overclocking.