The backup library option backs up MC's library, the database. It doesn't backup your media collection.
At the location your library backups are stored (I think RemyJ has that there for you, if you didn't change it) you can copy the latest file to your new pc and restore it, File, Library, Restore library. This will take care of all MC's settings, Views, etc etc.
The media files you will probably have to copy manually.