If you're not jailbroken, you don't really need your last backup. Just plug in to itunes on the new computer as soon as you get it and sync. New backup created.
Err... I have tried this and have lost all my media... (songs, photos, videos, and apps).... however the new backup does contain the SMS, contacts, calender, notes, and user settings.
The iPhone is a PITA to manage effectively...
If you copy your directory structure exactly from the old pc to the new pc (Music/iTunes and Documents/iTunes) you'll get your backups.
This works, but make sure to copy your songs and videos to the original directories-syncing can go awry if you don't.