Should I improve my old rig, or just start fresh?

Time for a new build. There is no viable upgrade path for that platform. Even an Intel i3 will serve you alot better.

Some games are graphics limited, and some are cpu limited.
I think GTAV is in the cpu limited category along with mmo aand strategy type games.

There is no good cpu upgrade for your motherboard if you need faster single thread core speeds.
Intel i3-6100 would be good and need a lga1151 motherboard.

If your games are graphics limited, then the psu capability comes into question.
Every psu will have data plate telling what the +12v amps or watts are. Look for that.
Your 6850 will run on a 450w psu and needs a single 6 pin pcie connector.
Today, a GTX1060 is considerably stronger and needs the same power.
Here is my stock approach on how to tell:
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To help clarify your CPU/GPU options, run these two tests:

a) Run YOUR games, but lower your resolution and eye candy.
If your FPS increases, it indicates that your cpu is strong enough to drive a better graphics configuration.
If your FPS stays the same, you are likely more cpu limited.

b) Limit your cpu, either by reducing the OC, or, in windows power management, limit the maximum cpu% to something like 70%.
Go to control panel/power options/change plan settings/change advanced power settings/processor power management/maximum processor state/
This will simulate what a lack of cpu power will do.
Conversely what a 30% improvement in core speed might do.

You should also experiment with removing one core. You can do this in the windows msconfig boot advanced options option. You will need to reboot for the change to take effect. Set the number of processors to less than you have.
This will tell you how sensitive your games are to the benefits of many cores.

If your FPS drops significantly, it is an indicator that your cpu is the limiting factor, and a cpu upgrade is in order.

It is possible that both tests are positive, indicating that you have a well balanced system, and both cpu and gpu need to be upgraded to get better gaming FPS.
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Agreed, that rig is essentially dead in the water at this point.
I would say save something like 600 or so to build something like this which is a small form factor PC with on board wifi.
Optimally though you want to get an i5 6500 with this, which has stronger cores and higher clock speeds, meaning it will stay relevant for longer, while the i3 is essentially at its maximum potential with the RX 480. the i5 mentioned is roughly $90 more on avg.
PCPartPicker part list: http://pcpartpicker.com/list/BxF29W
Price breakdown by merchant: http://pcpartpicker.com/list/BxF29W/by_merchant/

CPU: Intel Core i3-6100 3.7GHz Dual-Core Processor ($110.99 @ SuperBiiz)
Motherboard: ASRock H110M-ITX/ac Mini ITX LGA1151 Motherboard ($67.99 @ SuperBiiz)
Memory: G.Skill Ripjaws V Series 16GB (2 x 8GB) DDR4-2400 Memory ($64.99 @ Newegg)
Storage: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive ($47.49 @ OutletPC)
Video Card: Sapphire Radeon RX 480 4GB Video Card ($200.00)
Case: Cooler Master Elite 130 Mini ITX Tower Case ($42.99 @ SuperBiiz)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($43.99 @ SuperBiiz)
Total: $578.44
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-07-19 20:21 EDT-0400