Catalog and Picutre location

What version of Elements are you using?  I assume you are a Windows user.

Have you installed Elements on your new computer?  If so, restore the backup using that installation, and when given the options for restore location, choose a parent folder (e.g. Pictures) on your desired external drive.  Without knowing more about your folder structure on your old computer, it is difficult to make a recommendation for the options that are available. You either want to choose:

• New Location or
• New Location with the Restore Original Folder Structure

The first option will place all your folders in a flat hierarchy.  In other words, you will have a single level of folders under the chosen Pictures folder.  The second option will keep your existing hiearchy in place.  But since you may have duplicate hierarchy on your multiple drives, you may end up with a messy solution.  However, you can always clean up the hierarchy by click-and-draging folders to new locations in the My Folders panel of the Organizer.

For more detailed instructions see this Adobe HelpX Article.

Now, one final point:  If you restore the backup to your external drive, the Catalog folder will also end up there (I believe).  If that happens, you can simply cut/copy and paste the folder to the default location on your C drive (if that's what you want to do). The default location is:  C:\ProgramData\Adobe\Elements Organizer\Catalogs\