Something like this would be a decent upgrade :
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel - Core i5-7400 3GHz Quad-Core Processor ($177.89 @ OutletPC)
Total: $177.89
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2018-07-19 14:48 EDT-0400
It's twice the number of cores/threads and faster turbo speed which would help with gaming. The CPU is a little over the budget, but would be an improvement. I wouldn't recommend a second hand CPU, as you never know what has been done to them. But you may be able to find a decent processor around the 150 mark.
You will see a big benefit, not necessarily in FPS (but there will be an increase - for further FPS increase that's more GPU dependant), but in host processing, which will make gaming smoother as the CPU can handle more tasks simultaneously.