As long as you have the folder that holds the Calibre library saved, then you are all good.
The folder contains the file metadata.db and a lot of subfolders for the authors and books.
You just copy it all at once from the backup over to the new computer, and connect to it from inside calibre the first time you run calibre on the new computer.
More details here:
http://manual.calibre-ebook.com/faq....ter-to-another