Just be aware that dropping a new "core", that is, a motherboard/CPU foundation, in an existing Dell will require a Windows reinstall, as the Dell provided one won't work on a non-Dell motherboard. If that's a retail non-OEM copy you bought yourself, you should be good. You will also lose Dell support/warrant, as it won't be a "Dell" anymore.
You could just go the simpler CPU upgrade route, sticking with the low profile Radeon 5670 you already found.
Intel Pentium E5700 Wolfdale 3.0GHz ($70.99)
Intel Pentium E5800 Wolfdale 3.2GHz ($76.99)
They're not tri-cores, but it's less hassle and is drop in. In the end, outside stuff that needs over two cores, which will see a difference, they'll probably perform similarly otherwise, and the Intel will be less hassle (no motherboard swapping). You also won't need to account for another copy of Windows. Again, it's not a better alternative to the AMD one, just simply another option.
Dell PSUs are USUALLY pretty decent, but if that's a smaller size case with a lower wattage one, I'm betting it's not powerful. Then again, if it runs a GeForce 9500GT, I'd like to think a Radeon 5670 wouldn't be much more trouble. I'd check it to be sure.
Edit: Wattage isn't the be-all, end-all of PSUs. There's no "standards" when it comes to wattage ratings, so a 400W PSU of one make isn't equal to a 400W PSU of another make. Many "cheap" ones are just that. OEM ones don't have the top notch quality of aftermarket ones, but Dell PSUs, at least in the past, were usually okay enough.
Edit 2: Bonus for now. Have you considered BSEL pin modding that CPU? If the motherboard can support 1066MHz (likely, but look it up), you could overclock it by raising it's FSB. Google BSEL pin modding and you'll find alot on it. You could bump the FSB to 1066MHz netting you 2133MHz. If 1333MHz is possible (less likely), it'd be even more. It won't be as effective as upgrading but it's free for now.Edited by Princess Garnet - 3/27/11 at 8:42pm