Yes, move your catalog and previews folder to the external hard drive.
Thanks for the quick reply. Will I be able to use smart preview to ensure that I can still edit without my external hard drive? Or will smart preview files have to be located with the catalog files?
Lightroom is designed as a single-user, single-computer management system.
To keep multiple computers synchronized, you need to be manually copying the catalog (while not in use) from one computer to another.
The better approach is to put the catalog on an external drive and just move the drive from computer to computer, thus no manual syncing necessary.
Along with syncing the catalog, each computer needs access to the photos. This is easier than the catalog problem as the photos can be located anywhere (network drive or external drive). "Smart previews" can be used for much editing (not all editing, and no exporting) when the master images are not available.
The easiest solution is to put the catalog (and its preview files) on an external drive with the master images and just move the drive from computer to computer as needed.
Going along with what ManiacJoe has said, smart previews wouldn't be necessary in this case because both computers would have access to the master images. The good side of that is that you would be able to use Photoshop or other external editors when needed. Photoshop and other external editors cannot be used with smart previews. In the work scenario you are proposing, smart previews really wouldn't be necessary.