How can I find the CD Key used in a Diablo 2 installation?

The registration key is encrypted in one of the .mpq files in your installation directory, either patch_d2.mpq or d2data.mpq ... sources I found seemed unsure as to which. Not that it matters, because it's encrypted.

There are programs "out there" that will extract the key for you, but I would be very careful using them. I haven't used one myself, so I can't recommend any to you, and programs like that are sometimes located on sites that don't necessarily scan everything before it's posted. If you search for "Diablo II key grabber", you should get some hits that might point you in the right direction.

If it's already installed and registered on the other computer, most Blizzard games (Diablo 2 included) seem to respond well to just being flat-out copied from one computer to another. Assuming the CD key is stored in a file (can't find anything in the registry on a quick look-through), it would solve that problem and save you some time installing (and even patching).