How to make the most out of a one-on-one reading course with a faculty member?
Ask the professor about his/her expectations on meeting. A short meeting once a week is probably the norm (comment of Nate Eldridge). But ask. And try to schedule a regular time for it. I think that what is expected is that you would give a short report on what you learned and that you would ask questions. If the prof is being vigilant, you might need to field a few questions also, some that might be hard/impossible to answer. The intent will be more to get you to think in a certain way than to test you. 

But don't treat it as just "reading". Find a way to do some practice around the readings. If it is a well studied subject there will be exercises available and you can report on what you have done with those. In a more "state of the art" situation you will want to think about "what if..." to get a sense about where this subject might be extended. That would be for a quite advanced set of readings, of course.