richard k. means, the best progressive election lawyer in the state of illinois, explains that the procedure for obtaining a recount is quite clear:
A discovery recount is only available in Illinois to a candidate who is within 95% of the winner. Even then you get to recount only 25% of the precincts in every election jusridiction in the district for a pidling $10 per precinct. In order to change the result, you have to go to court and prove that, had certain very specificly described errors not been made, that your candidate would have won.
means is one of the authors of the 2002 handbook on illinois election law published by the illinois institute for continuing legal education and wrote the chapter on recounts in the state.