Mobo, RAM, PSU, and Case will all be DELL judging by his original post. He just wants to upgrade the CPU and everything else is stock, right? I would guess with that old board you are going to need a BIOS update at the very least. There is no guarantee that the mobo will support the new Celeron D anyway because those weren't around when your system was built. Has to support the proper bus speed, multiplier, and voltage settings for the Celeron D chip, and that is unlikely.
Check the latest bios updates from Dell for your system. They are downloadable from support.dell.com. Read the release notes to see if it mentions anything about supporting new processors or multiplier settings, etc.