When you're small (startup), hiring is a reflection of founders' values. When you grow big, it's a reflection of your company's values.
First hence, be clear about what your values are. It's not easy to figure those out. You have to interview several candidates and introspect after each interview, to narrow down on what you want. It's a bit unfair to first few candidates, but overall it works out better. Once you arrive at a nugget of values you can cohesively describe, you design your interview process and questions based on what you've arrived at.
Also realize that you cannot design direct questions to probe for values you want. You have to demonstrate that value yourself first and see if they get excited with it. e.g. if one of your values is transparency, you can't ask "are you transparent in your interaction with people"? You have to tell them some things about your company or the role that they are not expecting you to disclose and see if they value that.
If coming up with core values is too hard or too vague for you, then start with these three: Curiosity, Humility and Hard work. You won't go wrong with these. Most high performing people demonstrate these values.
In order to probe for these values, you can ask pretty much any reasonable question, and then pick up on cues. e.g. if it's a programmer interview, you can ask them a difficult coding question/assignment based on their background. Then see, if they ask good questions (curiosity)? Do they test their code (humility)? Do they give up, or continue to push through (hard work)? Don't expect precise answers to your question, but look for these signals.
When picking questions, you should prefer questions that are "peeling the onion" type questions. i.e. start with a simple question, let them answer it and then add constraints. Keep adding constraints/twisting until they are able to answer it. That will give you great insights into how they think and how they value.
Hope this helps.
[About me: Founder of http://InterviewKickstart.com. I've interviewed an obscene number of people in my career. Have been in the valley for a number of years. I have to think about this for a living]