When searching for any records, I use the following checklist that I created for myself:
- Learn what records might have been created in a particular time and place.
- Research which of those records might still exist, and which records are accessible to the public.
- Research what repositories might hold those records.
- Research which online repositories might hold those records.
To see what governmental records might exist for this time period, see the
FamilySearch Research Wiki's Article on Illinois Vital Records. It has an overview of what records are available, including a timeline of the introduction of statewide registration.
The section on Death Records gives several links for collections which can be searched online, and has a link to the article on Illinois counties if a search of the statewide records don't yield any results.
For example, Family Search has the name indexes Illinois, Cook County Deaths, 1878-1939, 1955-1994 and an index to Illinois Deaths and Stillbirths, 1916-1947 -- even though you are guessing your great-great-grandfather died before 1916, it would do no harm to search both indexes. Check the catalog entries for both databases, because they often contain notes about coverage issues that are not evident from the date range in the title of the database itself. There are also pages in the FamilySearch Research wiki for each collection, which have suggestions on what to do if you do not find any results in the collections:
For any collection at FamilySearch, these pages about the records can be accessed from the catalog entry by clicking the Learn more link, or by going to the Research Wiki and searching by topic or place.
It is tempting to leap directly to searching for a record that directly answers the question you want to know, but the difficulty with finding a death record is that people can die anywhere -- there's no guarantee that someone dies in or nearby the city where he lived most of his life, or that his body would be returned to be buried near other family members. Try to narrow the date bounds by working in small steps from the information you already have. One technique I use to narrow the date of death of a person is to gather the obituaries of all of their children and see if the parents are mentioned in the obituary as living or as having pre-deceased their children. If the deceased person has siblings, I do the same with the entire sibling set -- I try to establish the death order of all the siblings. Obituaries may not be accurate, but the residence clues for all the survivors gives you a pool of other localities to be searched; sometimes people died away from home while visiting relatives.
Newspapers are also a valuable source of information about when people are still living -- social news pages can list wedding anniversary parties, visits to relatives, and other everyday tidbits that demonstrate that the person you are looking for is still alive on that date.