Good low budget CPU

[Caiopixel]

I am looking for a good CPU which will not bottleneck newest GPUs. My motherboard is an H55H-CM (I know it's very old)

 

[RCFProd]

Time to perhaps get a new LGA 1151 mobo with a G4560 + B250M motherboard.

 

[atomicWAR]

Honestly I would upgrade your motherboard CPU and ram at this point. As old as your platform is finding a CPU will be difficult and likely not very cheap. However if you could find a cheap i7-880 it may be worth the upgrade depending on the CPU you have now.

 

[Ecky]

You can drop any LGA1156 CPU into that board, and it should work, though there's a possibility that the more power hungry ones (e.g. high-end i7's) might cause instability due to weak power circuitry on the board. I'd probably give it a try anyway.

Some CPUs to look at:
Core i7 860/870/880
Core i5 750/760

That said, even the highest-end i7 for your board is still going to be close to a decade old, and weak by today's standards. There is no CPU available for your board that will not cause slowdowns in any modern games.

 

[Crumpet 1]

Time to perhaps get a new LGA 1151 mobo with a G4560 + B250M motherboard.

I agree with this, or wait 2 more weeks for news on Ryzen 4 cores, looks like you can get an overclockable motherboard for only $70