(Click the character infographic to download.)
The Director is a little bit of a plot device, which is why there isn't too much to say about him. In the first few chapters, he's a great excuse for Huxley to talk all about the World State and basically ease us, as readers, into the world he's created. The Director's lecture to his students amounts to Huxley explaining to us what the deal is. After that, his whole horrible relationship with Linda is basically just the backstory to explain John's existence.
That said, the Director is still a real character. He's basically a typical, high-level Alpha who, unlike Bernard or Helmholtz, remains completely content with the status quo. However, like Henry Foster, he shows moments of humanity under the veneer of conditioned acceptance. Check out his conversation with Bernard in Chapter 6. When he talks about Linda, it's pretty clear that there were emotional, even monogamous elements to their relationship. Our suspicion is confirmed when the Director gets seriously defensive about this very fact, insisting that "it was all perfectly healthy and normal." That's when he goes all strict-teacher on Bernard, essentially threatening him with a lifetime worth of detentions (a.k.a. Iceland) if he doesn't shape up. That the Director is so upset over Bernard's seemingly emotional attachment to Lenina is very likely because the Director himself had such a relationship with Linda.