Chicago has a shortage of apartments that are considered "affordable" - costing less than 30% of the average individual's income. A recent report from Depaul discovered that the median cost of a 2 bed in Cook County was $1000, which would be "affordable" only if you're earning about 40k per year. We're pretty expensive here. The average rent list price in Chicago for a 1 bed right now is around $1300 according to Zillow, so while you may be able to find something at $600, you get what you pay for.
I'm seeing some 1 beds on the south side for $550 to $650. You' have to go south of 35th street to find them. Neighborhoods like Englewood, South Shore, New City and Grand Boulevard have a selection of them at that price range. I haven't done any client searches down there in a long time though, so I can't speak to the quality of the apartments.
On the north side I've done a few searches for clients this year who were looking for 1 beds in the $600-800 range and the pickings were very slim. They do exist - I know of a north side 1 bed coming up in the near future that will probably list at $680 with heat included - but they are few and far between. Most of them on the north side that I've seen in that range are junior 1 beds, generally 600 square feet or less with a living room, a galley kitchenette and a petite bedroom.
In all cases when renting in the city I do recommend doing your homework about the neighborhoods in question using resources like Wikipedia, Walkscore, Everyblock and Trulia's own crime map. If possible, you should definitely make an in-person visit any area both during the day and at night where you're planning to live before signing a lease. Neighborhoods vary drastically in character and even on a block-by-block basis.
Renting a house anywhere, city or suburbs, is going to be more expensive than renting in a large building with many apartments.
Regardless, I'd say a more realistic budget would be $900-1000 if you want something in a pleasant area, large enough to comfortably house 2 people and with a reputable landlord that isn't in or near foreclosure.