How do I get the private categories to display?
Private categories work in conjunction with user-groups. You can create a user-group and assign some private categories to that group. Now, all users (KB Members) of that group would be able to view those private categories after login on the front-end. Please have a look at the screencast below to see how private categories and user-groups work in PHPKB software.
This video would explain to you how you can associate private categories with user-groups for group-based access to the private content in your knowledge base.