The main element to get FPS is the GPU, the CPU is auxiliary so that the GPU can realize its full potential and also has a role in streaming. The best CPU in the world does not push the GPU beyond its potential.
The FPS number depends on each game. Your GPU in some games will get 60 and in others possibly not. What is certain is that your current processor is weak for the GPU and is limiting it.
You can put a sixth generation Intel or a seventh generation Intel without changing the motherboard (although this will surely require upgrading the bios of your motherboard). The minimum that I would advise would be an I5-6400 (6th) or an I5-7400 (7th).
The problem is that both are overpriced for what they give (even in the second-hand market) and exceed their budget.