New CPUs don't use DDR3 anymore, so you'd need to go with an older one to be able to reuse your RAM. Be aware that 1.65v RAM is not recommended for Intel 4th generation "Haswell" CPUs, so if yours is 1.65v and not 1.5v it's probably a good idea to replace it anyway. Intel's newer 6th generation "Skylake" CPUs are faster, draw less power, run cooler, and have newer chipsets with more modern connectivity, and use DDR4. They're also not discontinued like Haswell chips.
IMO, here are your options:
1)
Haswell i5 4xxx (cheapest you can find)
H81 motherboard
OR
2)
Skylake i3 6100 OR i5 6400/6500
H110 motherboard
8GB DDR4, sell your DDR3
The i3 6100 is $120 new with warranty on Newegg/Amazon and is probably faster than lower end Haswell i5's in most cases. The i5 4460 runs about $150 on eBay. The i5 6400 can be found for about $160 on eBay, and a motherboard will run you ~$40-50.
If you sell your current RAM you can probably get DDR4 for about the same price, so that's worth considering.