How to ask someone (outside the department) to be on the thesis committee
From my experience, having cross-department committee members is very common, and can actually be very beneficial to the student, as you gain access to expertise and perspective you might otherwise miss. There's usually at least some connection between the research being done and the expertise of the faculty member. To use myself as an example, my research involved behavioral neuroscience, but I had both an electrical engineering faculty with signal processing experience and a psychology faculty on my committee, and they both gave very useful advice with my project. That being said, I had a committee member who was completely unrelated to my work (my focus shifted midway through my PhD career), and he requested to still stay on despite the lack of direct research connection; he was happy just to give advice and provide feedback.

You can find potential committee members by talking to other graduate students, and by looking at what theses professors have worked on (as you suggest above). I definitely recommend talking to other students; some professors are more difficult to work with than others, and you likely won't get that type of information from your advisor.

Regarding how to go about talking to the professor, I would just send a simple email asking for a meeting about this topic. If they're simply not interested, they'll say so, and you'll save yourself time. If they are, meet to discuss the following:


who you work for
what you're researching (both the ten-year research goal, if there is one, as well as your specific project)
why you think this faculty member can help you
why this person should be interested (this should be tacit; i.e., you shouldn't have to say it explicitly, he should understand it from the above items)