The stay
There are tales of pros sleeping next to their bikes during the Tour de France. There are also tales of travelers doing the exact same thing when under way on a trip - for security and pre-ride tune ups. The best way to visit Chicago, as you'll read below, is to bring a bike along - either ship it or check it in as luggage.
Checking it in?
If you have a good bike case and you don't mind checking your luggage through and, with certain airlines, paying the extra baggage fee, then it's always ideal to have your own bike with you. Air Alaska flies into and out of Chicago, and they have the lowest bike luggage fee I've seen, at $70. If you're lucky enough to live in one of their destination cities, they're your cheapest option.
Once you're at O'hare or Midway airport, the taxi stand attendants will order you up a van to fit your luggage - there's no additional fee for the service, and you don't need to tip them. You can also order up an Uber XL at both airports, and they'll scoop you up at the baggage claim.
Shipping your bike
Chances are really good that shipping your bike is actually less expensive than the cost to bring it with you on an airplane. BikeFlights.com gets excellent shipping rates with FedEx, though you'll need a few days before and after your trip for shipping because it travels with FedEx ground (3-4 days each way). If you have a hard case, pack up your bike and schedule a pick up. If you point to point packing, call an independent shop and they'll usually pack up your bike for $40-$50 bucks - the folks at Turin Bicycle on Chicago's north side do an awesome job, and they're a great connection for local group rides as well.
Renting a bike in Chicago
Honesty is the best policy, and to be brutally honest, there aren't too many good rental options in Chicago if you're looking for a high end road bike. You'll find a few very touristy, high-volume rental businesses on the lakefront (Bobby's Bike Hike, for example), and a couple of local shops, such as Mox (close to downtown), and On The Route (two locations on the city's north side). There are some good options on Spinlister too, and if you're a decent human, you'll find it's not uncommon for independent shops to offer rentals under the table if you call them and ask for some help finding a bike.
Where to stay
There are plenty of hotels in Chicago that'll let you keep your bike in your room. There are even more that'll park it for you in storage. Choose the prior - you'll have it safe in your room with you, and you can tune it before the ride, and clean it afterward and maybe even snuggle up with it and watch a movie at night.
Hotels that have confirmed "It's ok to keep a bike in your room with you, weirdo:"
An excellent resource for bike-friendly hotels in Chicago and many other U.S. cities is at a great site called BikeAbout.com.