Advice on upgrading a Pre-Built

Without having to pretty much entirely replace the thing... I'd say either an FX 6350/8320 + GTX 1060. I have some reservations with recommending an 8xxx series CPU as the motherboard in that machine is likely a cheap-out motherboard, based on the fact that it is a pre built, and that it is a cheap pre built. You would likely be able to run it, but there is absolutely no way that you would be able to overclock it whatsoever.

The other option would be to go either to Skylake or Haswell, which would end up costing more money, but would essentially be replacing the entire machine.

PCPartPicker part list: http://pcpartpicker.com/list/9QDMVY
Price breakdown by merchant: http://pcpartpicker.com/list/9QDMVY/by_merchant/

CPU: Intel Core i5-4590 3.3GHz Quad-Core Processor ($189.99 @ SuperBiiz)
Motherboard: Gigabyte GA-B85M-DS3H-A Micro ATX LGA1150 Motherboard ($55.98 @ Newegg)
Video Card: MSI GeForce GTX 1060 6GB 6GB 6GT OC Video Card ($249.00 @ B&H)
Total: $494.97
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-11-08 01:46 EST-0500

Haswell i5 w/ a motherboard that's probably just as horrible as the one that's currently in your machine, but fits your needs. With the non k i5, you don't need to overclock. You just need the proper slots on the thing to put all of the necessary components in. Which this has.

With Haswell you can reuse your current set of DDR3.

PCPartPicker part list: http://pcpartpicker.com/list/BFxfhq
Price breakdown by merchant: http://pcpartpicker.com/list/BFxfhq/by_merchant/

CPU: Intel Core i5-6500 3.2GHz Quad-Core Processor ($194.89 @ OutletPC)
Motherboard: MSI B150M BAZOOKA Micro ATX LGA1151 Motherboard ($57.98 @ Newegg)
Memory: Corsair Vengeance LPX 16GB (2 x 8GB) DDR4-2400 Memory ($74.99 @ Newegg)
Video Card: MSI GeForce GTX 1060 6GB 6GB 6GT OC Video Card ($249.00 @ B&H)
Total: $576.86
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-11-08 01:48 EST-0500

Skylake, again with a poor motherboard, but the same goes as with the Haswell one--doesn't need to be super fancy. You need a new set of RAM with Skylake, so that's an added cost. I threw in 16GB instead of 8GB as it was still within your price range. You can easily add another set of the same RAM that you currently have in your machine to expand it. Just make sure it's the exact same set.

Took another look at, the 8xxx series has the same TDP as the 6350. I forgot that that's what they did to the 6300 to make the 6350. Just... gave it more power to clock it up. So, in lieu of the 6350 I'll give a variant with the 6300 which is a lower TDP CPU and one with an 8350.

PCPartPicker part list: http://pcpartpicker.com/list/kgz29W
Price breakdown by merchant: http://pcpartpicker.com/list/kgz29W/by_merchant/

CPU: AMD FX-8350 4.0GHz 8-Core Processor ($149.00 @ NCIX US)
Video Card: MSI GeForce GTX 1060 6GB 6GB 6GT OC Video Card ($249.00 @ B&H)
Total: $398.00
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-11-08 01:52 EST-0500

PCPartPicker part list: http://pcpartpicker.com/list/t9dGqk
Price breakdown by merchant: http://pcpartpicker.com/list/t9dGqk/by_merchant/

CPU: AMD FX-6300 3.5GHz 6-Core Processor ($94.89 @ OutletPC)
Video Card: MSI GeForce GTX 1060 6GB 6GB 6GT OC Video Card ($249.00 @ B&H)
Total: $343.89
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-11-08 01:52 EST-0500

Note about the cpu: if GTA:V is one of the games you are aiming for i'd pick a cpu better than the FX6300, i used to have an FX6350 and GTA:V was one of the games i was being cpu limited in and triggered me eventually upgrading. The intel options are a lot faster.