Hmm, I wouldn't ask them how they think your skills and qualifications would fit in. Honestly, it's your job to tell them that in the interview, and if you haven't done that by the time they get to asking you if you have any questions, you're not in good shape.

I like to use the questions period to show them I'm interested in their organization/firm, to show that I've done my research, and to get an idea if I'd like to work for them.

After they say "do you have any questions for us?" I usually say something like "I've looked at your website and done some of my own research on you, so I've got an idea of the basics, but I do have a couple questions."

Here are some questions I ask (that usually go over well):

1) Why did you join this firm? (Let's them talk about themselves)

2) Can you give me an example of a typical project that I'd be working on this summer? Or one you gave an intern last summer?

3) What types of things do you look for in a summer intern? (If you're feeling bold, after they answer say "We like, X, Y, and Z", you could casually point out that "Oh my gosh, that's crazy, because I have X, Y, and Z.")

4) Ask about specific projects or aspects of the organization that they would be proud of. For example, if it's the ACLU, you could say "I saw that you just represented the KKK against the City of St. Louis. What was it like working on that case?"

5) I always end by asking "So what are the next steps in your hiring process?" I think that shows that you're interested in their organization, and want them to follow up with you. At the very least it will help you think of a timeline. At the best, they'll say something like "Well, we're going to hire someone in 3 weeks, but if you get an offer before then, let us know and we'll try to get back to you quicker."