It sounds like what you want is to be able to use lightroom and your images on different machines and have them sync by having all of your image files on an external drive and your Lightroom catalog in Dropbox, then...

You can simply copy your Lightroom catalog file (someFileName.lrcat) into your dropbox folder. If you then open Lightroom by holding down the option key when you click on the Lightroom icon, it will ask you for a catalog file to open. Navigate to and open the catalog file now located in your dropbox folder.

This is all that is really necessary. As long as you make sure that you close Lightroom on one machine before opening Lightroom on the other, you can now move your external drive between machines and open Lightroom on either one to edit the files. You should have the folder path to the external drive the same on each machine, but, if you can't, it's pretty quick and easy to find the files again in Lightroom.

As long as you close Lightroom on one machine before opening it on the other, all of your edits will be available on each machine.

A more detailed explanations is available at: http://www.diyphotography.net/dropbox-lightroom-catalog-sync-set/