The Deus Ex version of the Illuminati starts roughly the same time as the real world group. It was founded as a group of philosophically minded individuals in 1776 by law professor Adam Weishaupt. One member does claim that certain aspects of the Illuminati, such as its financiers (the Knights Templars) predate 1776 but not the Illuminati itself. Using this financial center they slowly begin to control the growing banking industry, leading up to dominance of a centralized banking system.
A quick digression: the Knights Templar, or the Poor Fellow-Soldiers of Christ and of the Temple of Solomon, were a chivalrous order originating during the Crusades who eventually became quite wealthy and powerful, as they were some of the earliest examples of international bankers. The order was destroyed by Phillip II of France with the help of his chosen pope because of the massive debts Phillip owed the Templars, but various legends and conspiracies state they continued on in secret. A very nice little nod to history, there. Interestingly, the Illuminati has been accused of running the French Revolution, which destroyed the French monarchy. Revenge?