Abby, the main reason for backing up your catalog is insurance against a hard disk failure. For this reason it makes little sense for the backup to be on the same drive as the working catalog. When you close LR and the window asking if you want to do a backup comes up, there is a "Browse" button with which you can choose a location on the external drive.
Also, it is unclear whether you shoot and edit Raws or jpgs, but if you shoot Raw there is no need to permanently store jpgs. As long as you have the Raws and the catalog, you can in seconds make a jpg when you need it, usually for sending off to somewhere else - web sharing or printing - and after it has been sent it can be deleted.