Best i5, mobo, and ram for 250-270 dollars?

Solution
Skylake. The 6400 is a little cheaper but you only save around $15 and if more than one core is fully loaded it won't turbo as fast as the 6500 at stock. May as well pay the extra $15 in my opinion.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor ($194.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B150M-D3H Micro ATX LGA1151 Motherboard ($67.98 @ Newegg)
Memory: G.Skill Ripjaws V Series 8GB (2 x 4GB) DDR4-2133 Memory ($28.99 @ Newegg)
Total: $291.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-03 21:21 EDT-0400

If the 6500 is out of your price range...
Skylake. The 6400 is a little cheaper but you only save around $15 and if more than one core is fully loaded it won't turbo as fast as the 6500 at stock. May as well pay the extra $15 in my opinion.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor ($194.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B150M-D3H Micro ATX LGA1151 Motherboard ($67.98 @ Newegg)
Memory: G.Skill Ripjaws V Series 8GB (2 x 4GB) DDR4-2133 Memory ($28.99 @ Newegg)
Total: $291.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-03 21:21 EDT-0400

If the 6500 is out of your price range, there's the 4th gen i5's.

PCPartPicker part list / Price breakdown by merchant

CPU: Intel Core i5-4460 3.2GHz Quad-Core Processor ($169.99 @ SuperBiiz)
Motherboard: MSI H81M-P33 Micro ATX LGA1150 Motherboard ($43.88 @ OutletPC)
Memory: G.Skill Ripjaws X Series 8GB (2 x 4GB) DDR3-1600 Memory ($34.87 @ NCIX US)
Total: $248.74
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-06-03 21:26 EDT-0400

I can't really speak for either of these motherboards, they should work ok but keep in mind they may not have a lot of options like number of fan headers, maximum ram capability (16gb vs 32gb on 1150), features like that. They're really low budget boards.
 
Solution