Who are your teachers? The staff of some schools skews toward young teachers fresh out of university; others have more career teachers with accompanying families. Both are fine. It just depends which you prefer (and it might say something about the benefits and relocation package offered by the school to its employees). Also in this category: what is typical teacher turnover? What nationality do the teachers tend to be? What qualifications or experience must they have? Etc.
Who are your students? In my own research three years ago, I came across schools where if I had sent Miriam there, she would have been the ONLY non-Muslim, or the ONLY American, or the ONLY blond-haired child, etc. Of course none of those are deal-breakers all on their own, but you should ask this question so that you aren't surprised on the first day of school that your child is the only non-Muslim (or whatever) there!
Can I see your library? What are the lending privileges? In many foreign countries, your child's school library may be the only library you have access to. So try to find a good one if that is important to you! Some school libraries may give borrowing privileges to younger siblings, or allow students to check out more books with a parent card.
How big is your school? Do you have multiple classes per grade level? What is the highest grade level offered? Are more grades being added each year?
What kind of curriculum do you have? At English-medium international schools, it tends to be one of the Big Three (US, UK, or Australian). You can also ask the school what kind of re-entry your child would have in a school in their home country - does the curriculum transfer straight across? How about exams at higher levels (IB, AP, GCSE, etc.) - are they offered and/or accepted in the home country?
What are your admissions requirements for students? This is a tricky one. Some schools require read/write literacy in English before a child can enter Year 2. But there are a million exceptions - if an older sibling has already been accepted to the school; if the child is of a certain age (you can't send a 10-year-old to Year 2); if what the heck, you can sign the tuition check so come on in; etc. You should also ask the school if certain nationalities (as determined by passport) are excluded from admission.
Are the children ever segregated by gender? Some schools segregate after a certain age. Some have co-ed classes but segregate for PE swimming sessions. Some have entirely different campuses for girls and boys.
What is the classroom atmosphere? Do teachers keep students in line by yelling, or ringing a huge cowbell, or by shaming troublemakers (all of which I've heard of here)? Make sure you find an atmosphere in which your child will be comfortable.
Does the school have an online presence? This may be a simple website, or it may include a more extensive parent/school communication system with homework postings, announcements, calendar, etc.
What are the school fees? There is tuition, of course, but don't forget about uniforms, books, transportation, registration, and activity fees. If your employer is sponsoring tuition, find out if that includes all the extras (it probably doesn't).
One last important factor in choosing an international school is how you feel about it overall. I remember walking into the girls' school for the first time three years ago and more than any other school I had visited, that school felt like home to me. A school could sound really good to you on paper, and have all the right answers to the above questions, but still not be a good match for your family. Make sure you let your intuition have input in this decision as well!
Did I miss anything?