In general, there are two paths to do this type of research: bricks & mortar, and people. Bricks & mortar deals with the physical structure of the building -- the architectural style, the architect, the contractors, alterations through the years. People encompass the owners who previously owned a business or lived in the building.
A helpful resource to get started is a publication from the city's landmarks commission
Another easy way to begin your research is to search the building's address in the digitized Chicago Tribune. The database is available at the Newberry Library (in-library use only) and also on the Chicago Public Library site (remote use with your CPL library card). There may be references to real estate transactions, people, or advertisements related to the business/building.
City and business directories can also be useful tools in your search. These are available at the Newberry Library as well as the Chicago History Museum Research Center. Some years have been digitized and available through the Newberry's Chicago Ancestor's site
Building permits are another good resource City of Chicago building permits (1872-1954) are available on microfilm at the Chicago History Museum and the University of IL Chicago. The University of IL at Chicago has digitized copies of the microfilm
