What skills, work experience or educational background do you look for when you recruit for these jobs?

Medical schools have a specific set of prerequisite courses required for admission so most applicants will come from biology, chemistry, biomedical engineering, and other sciences majors; however I have met literature majors, theater majors, and history majors during my time working for a medical school. As long as you have the grades and test scores to indicate your academic strengths, you are off to a good start.
In addition to grades, you will want to get hands on experience. Whether you are volunteering in a hospital and transporting patients, coloring with children in a pediatric unit, or answering calls as a receptionist - you will want to have an understanding of what it feels like to be in this profession. You may also want to interview a doctor or someone else in the medical field to get a "day-in-the-life" understanding and learn some do's and don't's

A lot of times in the medical field it is not so much the educational background as it is what you have obtained certification, registry, or licenses in.

Figure out what you want to do. Look at the credentials needed, than choose the education that will prepare you to be eligible for, and pass, the boards for.

Also, any actual experience is again more important than just an education. The education is the starting point but where you do your live training and how much hands on experience you get is key.

Choose the education based on where they will send you to train and what kind of track record they have for students passing boards. That's what I look at when considering someone. I also always bring them in to see what they can do in real life so, again, the real life experience is key