Family Search has made available online an index of burials in the Archdiocese of Chicago Cemeteries. The index was prepared from microfilmed records made in 1989. In that regard, the index does not include records beyond 1989. If you find someone of interest in the index, you can click on the name and be taken to the image of that record. You will need to be logged-in at familysearch.org to see the image. The images are of the index cards that reside in the cemetery offices. While I did not try alternative spellings, I did not find records of ancestors known to have been buried before about 1912 suggesting the index card system wasn't used in the earliest days of cemetery records. Besides the name, the date of death, date of burial, grave location, and address at time of death might be given, but not always.
Some of the cemeteries have kiosks at which you can conduct a search which gives the name, date of burial, and grave location (but no home addresses). This index is updated with new burials to keep it current. Unfortunately, that index is not accessible online.