The quad core phenom with 8 gigs is fine. Let's get the "bigger" graphics card working and decide from there if the cpu needs to be upgraded.
Couple questions:
1. Which graphics card did you choose?
2. Why do you believe you need a more powerful PSU (not enough connectors, blue screens, no boot, random shutdowns?)
3. What is the current Dell PSU rated at (will be listed in Watts on a sticker somewhere on the PSU).
4. What's the model # of the Dell?
If you picked a beefy graphics card ($250+), it will most likely need a higher wattage PSU. Cheaper cards may not.
With all that, if the graphics card you got definitely needs a bigger PSU, cheapest route is get a new ATX Power Supply, and a Dell/ATX PSU adapter, something like this:
http://www.directron.com/hp24mini.html?gsear=1
But, in my opinion, the "cleanest" way to go is to buy a bigger PSU and a new AM3 Motherboard, trash the proprietary Dell motherboard, and just swap the CPU/Ram over to that.