How do I know if my CPU is bottlenecking my graphics card?

Your current CPU will not bottleneck the 5970 even at stock speeds.

Pretty easy way, start a game up, wack everything on low/medium note down the FPS (Use fraps to show FPS), then put everything on High + AA and AF to the max and note the FPS, If your FPS results on low/medium are very similar or the same as the FPS on High + AA and AF then you have a CPU bottleneck

Alternatively you could just change the resolution say from 800 x 600 (note the FPS) and then to your monitors max resolution and note the FPS, then compare the results, remember to disable V-sync.

Install ATI Tray Tools. Configure the onscreen display to show the CPU Load and GPU Activity. Then, go the the graphing module and select the same items in the graphing module (and activate logging if desired).

Play your games as usual. The onscreen display will indicate at any moment which of the two are maxed out and you can review the graphs and logs afterward to determine general trends.