As a public servant, after serving a client, he begins to talk about his personal life, how would I handle this situation?. Do I excuse myself and say "I am busy, I need to get back to work", do I listen but not answer, or converse to allow the conversation to carry on longer or participate in his conversation for as long as it takes?

Respect his time and yours. First and foremost. Don't blow him off. But don't hold a 2 hour conversation. You accomplish this as follows.

Him: You know, my daughter is dating this jerk, and I tell ya; I think he's bad news.
You: Your daughter sounds a great girl, have a great day sir.

OR

You: That's unfortunate, is there anything else I can help you with?

What is your specific job title, if I may ask? I deal with this on a daily basis, fortunately I have time to converse a bit, and am encouraged to to get to know the patient and their hobbies and uses for their glasses better. I love getting to know our patients. I've learned that if you listen they think you care and if they think you care, they want to return because they are familiar with you, and feel you respect them.