votemegatron :
I did as you suggested and ran a log; here is an IMGUR gallery, please be aware I am not completely savvy with this, so I just included...everything. Picture heavy.
This is with GTA V at 1080p, everything maxed, AA/FXAA off, grass on very high, with the advanced render set to 50%. I was barreling down the city in the fastest car, and only once did I catch a FPS drop (60-57). I just noticed CPU runs higher now in some games and got concerned.
http://imgur.com/a/7UcHE
You don't need to go that far. Sorry, thats on me, I should have explained it better. Ok, download and install HWMonitor and CPU-Z from CPUID. They are both good to have.
Open your CPU-Z and take a look at whats there just out of sheer curiosity, and perhaps something may jump out at you. Then close it, open HWMonitor and let it run. Run your game, and you should have a screen that looks something like this:
http://imgur.com/R4seZWM
This is mine while under medium / light load, specifically, roaming in GW2 WVW, no mass or large battles. On this picture, I highlited with red what you should be looking at, which is your CPU load and your GPU load. If your GPU still has room to spare while your CPU is all maxxed out or close to it, then its the CPU that is the bottleneck.
Also, the 1st 3 which I had to collapse for this picture, is the mobo. you have your DRAM voltages, temps and fans, you can pretty much tell if its your RAM that is an issue as well moving data. Be advised, if your system gets overheated, it WILL throttle, on everything, thus cause lag. You need to find out what the expected idle and heavy load temperatures are for your components and compare that to the minimum and maximum values displayed here. I'm sorry but its been a long while since I have used those type of boards, so I don't remember.
Your solution could be as simple as adding an extra fan or 2, or as pricey and time consuming as buying whole new system for your GPU and everything in between.
Those graphs are a bit hard to read, but I see at one point you're hitting 100 deg on CPU as well as heavy voltages thats an issue. Pretty big one while your video card isnt stressed at all. Tone down a bit on the options and just alt-tab out of the game and see the max values, it will be easier to see and compare. I have an i5 6600K and so far under heaviest load it ran up to 66 degrees, most of the time it sits around 61-62 and idles around 30-33 which is a bit warm but I live in California so its consistant with the ambient value here. I know you should not be hitting 90s or approaching 100. You can run up to around 80ish and thats getting to warm, standard range is 60-70. So something is causing it to get that hot.
That said, i5 6500 should be OK for this, so it may be the cooling on it, or your air flow, or your main board. Also, you may try updating the BIOS on your main board, then go into it and set your cooling better, allow it to activate earlier then the defaults that are in there. And I'm not sure about turbo or any overclocking because that is not a K CPU. I never buy anything other then K designated CPUs when I go with Intel, so I just can't tell you what it should be. Or maybe you just need to clean it out, though if you bought a new GPU and installed it then you probably already did that part. Also, some people forget or dont use good thermal paste, it actually truly does make a huge difference for CPU cooling.