How are director levels increased?

Any way that you can gain experience, through rehearsal, competing in contests, and placing high in these contests (gaining medals for 1,2,3 places and making semis and finals in the last week of the season). I'm not sure, but upgrading staff, hosting events, and gaining accomplishments (look at my profile under my trophy case and you'll see I have a small number of small accomplishments) may give you more XP, but that might be a thing to ask a higher ranked person.

If you'd like, you can private message me with more questions if you'd like. :-)