I'm reading Patrick Lencioni's Overcoming the Five Dysfunctions of a Team right now. What he says about building trust with teammates is equally applicable to customers.
...it's best to start small. The point of the first exercise is to help people get comfortable with moderate vulnerability.
When dealing with customers, how can you keep it small and still get them to open up? I find that a great non-threatening personal question is. "Were you born here (or in this city)?" Notice that they can give a terse one-word answer and shut you down, and that's okay. (Let it go.) On the other hand, most people will open up a little. Then instead of nailing them with more nosy questions, you can share a little (just a little) about where you were born and/or how well you like the neighborhood. Boom! You just made the relationship more personal.