Ok I found this info on an outfitters website:

Illinois Non-Resident Archery Permits ($325) 1 Buck Tag & 1 Doe Tag
Illinois Non-Resident Firearm Permits ($200) 1 Either-Sex Tag
Illinois Non-Resident 5-day Hunting License ($28.75)
Illinois Non-Resident Full Season Hunting License ($50.75)
Illinois Habitat Stamp ($5.00)
It also says that archery permits will be available in July by Illinois DNR

I don't see anything about a drawing for a buck tag, but I think that is the way it would be operated.

You also MUST have a habitat stamp on the back of your license in order to hunt whitetails.

I should warn you though Illinois doesn't have any deer. Go to Iowa or Kansas

UPDATE: http://www.pikeland.com/statepermit.htmlThis explains a little bit about the process. BTW what part of Illinois are you planning on hunting?