You can install the software on both machines and move the catalog and the photos (or SmartPreviews if you like) along.
Use one external drive as the main catalog and photo drive. Move it from computer to computer as needed.
Use the second external drive as your emergency backup. Do backups of your catalog and photos frequently.
Using the Administrator disk management tools, make sure both computers are using the same drive letters for each external drive. (Like J or higher)