Anything close to the purple or red line that you recommend?
Probably gunna spend 3 or 4 days partying
I'm not sure if you've looked in Lincoln Park, but that might be an area you can look for a decently priced hotel. Lots of bars around that area near DePaul and Wrigleyville isn't far either. The red and brown lines run through there which you can take into the Loop.
The O'Hare suggestion isn't bad either, there's a casino near O'Hare in Des Plaines called Rivers Casino so there's that. I don't know much about the area there though but you'd definitely find a decently priced hotel and then you can take the Blue Line into the city like IU mentioned.
From the airport, the cheapest option is the subway.. Blue line for O'Hare.. Orange line for Midway. But if you don't want to do that, you can do Go Airport Express shuttle (which is what I use a lot) and there are coupons for it on the internet. Cabs are a last resort option IMO because of the cost unless you want the convenience of having your own cab. In that case, I'd say use this app called Uber (and theres a coupon for $10 off your first cab ride) which will give you that discount.
If you do decide to stay in one of the north areas like Lincoln Park, there's a decent hotel I stayed in when I first came here called
http://www.daysinnchicago.net/ -- Lincoln park also has a beach - North Avenue Beach so you'd be fairly close to that as well.
Also, try this website --
http://www.betterbidding.com/index.php?showforum=87 -- It's a site that people use to post what kind of hotel deal they got when they bid for one using Priceline's name your own price tool and you might be able to get a decent hotel in downtown for your price range. Just another idea, I've had success with it as well.
Enjoy and hope this helps!