I have set up a private forum and set it up to show the login dialog when anyone enters the site. I would like to put the link to register on that login dialog to make it easier for people to find.

Gisbill,

I'm pretty certain that barring any modifications of your own and as long as registrations are enabled the login page should contain a link to register near the top left of the login page. (At least with the So Clean theme) It is possible other themes may have it located elsewhere, but double check the login page again and make sure registrations are enabled for the forum.