Running GTA V at highest settings, over 60FPS.

A gtx 960 will not be able to meet those requirements. Your cpu is also to weak to not bottleneck a gtx 960. What is your psu as that needs to be decent otherwise you won't be able to upgrade. Also having more ram doesn't hurt.

Check systemrequirementslab.com. Have it scan your current config to have it identify which one (CPU, GPU, or RAM) is going to cause the biggest issues.

As far as upgrading:
-- Adding RAM is the cheapest, but at 8 GB you don't need more for GTA V.
-- Upgrading your CPU is going to help quite a bit. Your current CPU is 4th-tier (http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html), but although the AMD chips only top out at 2nd-tier, that a 2-tier jump (which is usually the minimum you want to do when upgrading). Right now, Tom's Hardware really recommends the FX-8300 in the "$100-200" range (http://www.tomshardware.com/reviews/best-cpus,3986.html), & it's apparently running about $113 on Amazon right now, but any of the FX-83xx chips would be good updates for you. Unless your board is limited to 95W chips, though, I would avoid the FX-8320e & 8370e. However, chances are good that your CPU would be "OK" for GTA V.
-- Your current GPU is almost literally bottom-of-the-barrel, coming in around 35th-tier on Tom's list (http://www.tomshardware.com/reviews/gpu-hierarchy,4388.html), so it's going to be a) the most necessary upgrade, & b) the most expensive part. Having said that, the GPU you pick is going to depend on your resolution. For example, if your monitor is matching the rest of this system & is only capable of hitting 720p resolutions, then even a GTX 960 is going to be overkill. At that resolution, you'd be better off with Tom's recommendation of AMD's R7 360 (http://www.tomshardware.com/reviews/best-gpus,4380.html): it'll match the hardware better, plus it will be a lot cheaper (running about $120 USD). If your monitor can hit 1080p resolutions, then a GTX 960 will be "OK", as it runs in between the "playable" GTX 950 & the "maxed-out" GTX 970 recommendations.

If your budget is going to be $200 USD or less, try upgrading your GPU to one of nVidia's newer GTX 950 models. They've come out with some that, like the older GTX 750/750Ti, only use the PCIe slot for power (i.e. no PCIe power connectors needed). If your budget can stretch farther, though (especially in the $300-400) range, then consider adding the FX-8300 to your GTX 960 (or again, if your PSU is a bit iffy, one of the GTX 950 models). If you can add another $100 or so on top of that (i.e. $500-600 budget), then consider a) upgrading your PSU (you can easily get one of EVGA's SuperNOVA B2 or G2 750W units for around $100), b) adding that CPU, & c) if there's still cash maybe getting a GTX 970, or even an R9 380/380X -- note quite as spiffy as the 970, but much cheaper & also a step up from the 960.