Good Morning Annette-
I'm afraid I can't tell you much about the Hamilton School/College of Law in Chicago.
From what I can tell, it operated in the late 19th century and early 20th centuries (perhaps ca. 1890's-1920). Most of the references I could find indicate that it was a correspondence / "by mail" school. I don't know what happened to it after ca 1920, but my guess would be that it completely closed down.

As for archives....
As a practicing attorney in Chicago, I am quite sure that your ancestor was probably a member of the Chicago Bar Association (founded 1874). I don't know what kind of archives they maintain, but given lawyers' love of records, I would think they probably do maintain a membership list going back to the founding of the organization. Might be worth a call or letter to them. Perhaps they might also be able to supply some info on Hamilton:
Chicago Bar Association
321 S. Plymouth Court
Chicago, IL 60604
tx (312) 554-2000
fax (312) 554-2054
info@chicagobar.org

Might also want to try the Illinois Bar Association. Don't know when they began operation, but I would think they probably go back to the time period you are interested in:
Illinois State Bar Association
424 S. Second St.
Springfield, IL 62701
tx (217) 525-1760
fax (217) 525-0712
info@isba.org

Hope something here helps in your search-
Paul