I game on a 1080p/60Hz monitor (running dual) with a i7-3770k and gtx970 and there's nothing I play that doesn't land me over 60fps, even the use of highly modded, enb skyrim or Metal Gear 5 at a 4k DSR.
Upgrading isn't going to change that. On a 60Hz monitor, you get 60fps. That's it. Doesn't matter if the cpu/gpu can output 150fps or 300fps, you'll get the same details, same resolution, same 60fps output.
Doesn't matter if you drive a mini-van, Honda civic, Ferrari or formula 1 race car when the speed limit is 55mph. All those cars will sit side by side on the highway, at 55mph. Anything after that, top speeds, acceleration, whatever, is absolutely wasted potential.
You could slap a RTX2080ti on that cpu, and get exactly the same results as you get now. The only way to justify the change would be a reason, like needing ultra settings at 1440p/144Hz or 4k/120Hz etc where gpu power is necessary to get the fps on-screen that the cpu can deliver.
Unless you aren't getting 60fps, there's no reason to upgrade/side-grade. If you aren't, then figure out why not. If changing graphics settings changes fps, the gpu is the issue holding you back. If changing details doesn't really change fps, then it's the cpu capping out. Cpu sets fps limit. Gpu has to live up to it.
Either way, you'll answer your own question. You'll either benefit from gpu upgrade or benefit from cpu upgrade, whatever is the reason for lack of fps.