Congrats on U of C! I did residency there and was very happy. One of the things that is great about U of C was that there was an emphasis (and enough flexibility in the workflow to support an emphasis) on parts of your career other than the clinical (as important as that is). Some, like myself, did research. Others got involved in ASA at pretty high levels (one was president of the resident component).
As far as deanships go, there are a LOT of different deanships and administrative positions within medical schools, from proper DEANs of students or of medical schools, to all these minor deanships, to ombudsmen, program directors, admissions committees, GME leadership, IRB members/chairs, etc. I think people who become higher-level deans DO have special qualifications and experience, but there are a lot of other positions, as above. The thing I'd encourage you to do is to decide that this is your path, and get involved early and seek out mentorship. At U of C, get on the GMEC committee and volunteer for program reviews. Seek the mentorship of Jerry Klafta, who is the medical school's new ombudsman and others in the department (like Bill McDade) who have other administrative posts.
What I've noticed is that people who get these positions didn't just decide at some point they wanted them and applied for them. Their whole careers and many of their choices lead them to it, and they had people whispering their names in the ears of decision-makers. So, again, make choices that move you into those realms and seek the mentorship of those IN those positions or in other positions of influence.