What kinda computer? If your old computer is running (for instance) Windows XP, and your new one is running Windows Fista (heck, spit!), then:

a) For the sake of your sanity and to double its speed, install Windows XP on your new box! MS says you can, so it's perfectly legal and your new box will work like the old one, only faster.

b) The ghost of Digital Restriction Management[1] may rear its ugly head, and not believe that you have the right to play the music.

But the important parts are: finding the files and getting them across. I don't know how iTunes stores files (I don't use it), but getting them across is easiest if both your old and new computers have a network card. That way, you can send your files over the wire from one computer to the other. Maybe you can connect both computers to your Internet modem. Then you need to share the drives on your old box and access them from the others. If this kind of stuff scares you, get a friendly geek to help you.

The other way is to save your music onto a DVD and copy them that way. Or a USB key (mine is 4GB in size, and bigger ones exist).

While you're at it, see if you can export your music files to either the Ogg Vorbis or to the MP3 file format. They're less stroppy in the face of copying.

Good luck!

[1] Don't ever let anyone call it Digital RIGHTS management! The recording industry has no say over your rights.