Hi,
So much confusion...
Changing the CPU is going to end up very expensive. I would avoid that for now unless necessary. You'd need:
a) new CPU
b) new motherboard
c) new system memory
d) new Windows copy
*Yes, an i5-6600K will give a better frame rate, but the amount of CPU bottleneck varies a lot from game to game. Here's some examples:
1) Tomb Raider-> minor CPU bottlenecking:
http://www.techspot.com/review/1128-rise-of-the-tomb-raider-benchmarks/page5.html
2) Bioshock Ultimate->
http://www.guru3d.com/articles_pages/core_i5_6600k_processor_review_desktop_skylake,15.html
Some of them like Bioshock Ultimate can run well over 60FPS already. If you raise the resolution and/or graphics settings it uses proportionately more GPU processing but even if still CPU bottleneck it's not always a problem if the FPS is still good.
3) Other games will have higher CPU bottleneck, but you can still play around with the SETTINGS and have a great experience.
4) DX12 games will start coming out that don't have much CPU bottleneck with your FX-8350.
GPU performance:
https://www.techpowerup.com/reviews/NVIDIA/GeForce_GTX_1070/24.html
Ignoring the CPU bottleneck per game, this is still a really great guide. GTX1060 is arguably not worth the upgrade vs a GTX960 as it's probably about the perf of a GTX980. It's still up to roughly 1.7X the perf so that's up to you. I also like 8GB of VRAM for a high-end GPU now so for you the GTX1070 might be a good choice.
For games with minimal CPU bottleneck, the GTX1070 is up to roughly 270% faster than a GTX960. So you'll get somewhere between roughly 20% and 250% benefit depending on the game for average FPS increase.
So you might go from 30FPS to 60FPS whilst also having better visuals.
Summary:
If you were BUILDING a new system I'd say go Intel for sure. However, depending on the games you currently play spending the cash to upgrade may not be worth it.
*I suggest getting the GTX1070 (not a Founders card), then look at how your performance is for your games and see what benefits upgrading would bring.
**On the other hand, the GTX960 is still a great card so perhaps the question should be WHAT EXACTLY are you trying to achieve that you can't do now?
I'm still rocking a GTX680 +i7-3770K. I got all excited to upgrade to a GTX1080 but when I started looking at most of the games I play they are all 60FPS now at max or near-max quality settings. There are a few exceptions but I didn't feel I could justify the cost yet.
Other:
a) Tweaking is important. For example, some games I use ADAPTIVE VSYNC and tweak to maintain 60FPS (60Hz monitor).
b) MONITOR is important too. I really like 2560x1440, IPS. I don't have GSYNC yet, as I can't justify the cost (about $750USD for 2560x1440, IPS, GSYNC).
I play a variety of games, but some like CIV5 really look far better on a high-res monitor (and many of the "god" games aren't very demanding relative to some shooters).