I recently received a new video card as a birthday present. The card is an VisionTec ATI Radeon HD 4870 that requires a PCI Express x16 and 600 watt power supply.

First, my Dell Dimension XPS Gen 2 contains only a 420 watt power supply.
Second, there are no PCI Express slots (only AGP and PCI slots).

It is worth upgrading my current machine's motherboard and power supply?
(or)
Would it be cheaper to buy a new machine to support this new video card?

I'd like to use the machine both for games (like Crysis) and for home video editing.

Your computer is a proprietory manufacture make and model
Do not update the Motherboard to some new brand
The support page for your model tells you what it currently has (see here)

You are best to update the entire computer