Hi mschauer! I live in Chicago, and have my whole life. I've got no particular insight into DePaul, per se, but I do know a fair amount about renting in our fair city, and I'm happy to answer any questions as best I can.
If you've looked into this at all, I'm sure you've heard the statement "Chicago is a city of neighborhoods"--and it really is true! The city is full of neighborhoods that very in terms of safety/ethnic diversity/density/transportation etc and so on. I mostly know about the neighborhoods on the North Side, but that's OK because if you're studying at DePaul, the North Side would be most convenient for you. But the North Side is huuuuge, and we'll have to narrow it down a little further. A general rule of thumb for the North Side is that the further east you go (towards the lake), the more dense and urban the neighborhoods feel. There are some exceptions, but typically, along the lake front, you'll find more high rises, more walkable neighborhoods, and places where it's harder to park a car. DePaul is located in Lincoln Park, which is along the lake, and is a cool, urban feeling neighborhood. There's lots of bars, restaurants, nightlife, cultural stuff etc, and it's close to the zoo (like Bellefast said) and near the lakefront parks and bike paths. Very pretty buildings in that area, too, if architecture rings your bell. The neighborhood of Lincoln Park can get kinda pricey (for my tastes) and parking is a beast over there. I'd recommend that, if you're planning on living a little bit away from DePaul, that you try for a neighborhood along an El line, and commute by train.
DePaul is closest to two train lines; the Red Line and the Brown Line. The Red Line runs North-South through the city, pretty much close to the lake, so it goes through some of the denser North Side neighborhoods. You might want to look into neighborhoods along the Brown Line (which runs kinda northwest through the city from the loop downtown). Those neighborhoods still have a city feel, but you'd be more likely to find a place in a two- or three-flat (instead of a high-rise), maybe with a backyard, on quieter streets. In particular, I'd recommend you consider the following neighborhoods: Lincoln Square, North Center, Ravenswood (and Ravenswod Manor), Albany Park and North Park. Take a look at craigslist--there's a ton of apartment listings there, and you can search by neighborhood and limit your results by price, and you should be able to get an idea of what's available. There's plenty of apartment search services that will drive you around and show you places according to your specifications (the listing landlords pay a finder's fee for that service, but it's free for you and might be a good choice if/when you're actually looking for a place here).
Here is a neighborhood map that might help orient you http://www.wildonions.org/NeighborhoodsMap.htm
Some weird/unique things about renting in the city:
A lot of apartments try to schedule their leases so that they're available June 1st or October 1st. There's still plenty of options at other times of the year, but those are the times you'll find the most choice and competition. Not ideal for an academic schedule, I know! If you rent a place where the lease starts at a different time (like the 1st of August) the landlord might ask you to sign something other than a year long lease, like a 14 month one, to more closely coincide with that schedule. I don't know if most landlords consider this kind of thing a deal-breaker, but be prepared to answer that question.
Utilities: Water and garbage are included in the rent 99% of the time. 99% of the time, you'll have to pay for electricity. The big variable in utilities is heat--in some places the heat is included in the rent, and in some places it isn't, and that's often stated up front in the ad (if the ad says that tenets pay for "cooking gas" that means your heat is covered, but you'll have to pay the gas company for whatever your stove uses, usually <10$ a month). Most places are heated with gas (either forced air or radiators), and with our cold winters your gas bill can eat up a non-trivial amount of your budget, depending on how old or drafty your place is. Your landlord is required in Chicago to give you an estimate of what your heating costs will be along with the lease, and you can certainly ask for it earlier. That estimate will be based on what the gas company billed the previous tenant for. If you do end up taking a place in Chicago, you can enroll in the year-round budget plan with the gas company. They'll take the total estimated costs for the year, and then charge you 1/12 of that each month, which is great because it makes it much easier to budget because you don't end up paying 0$ in August and 200$ in January.
Chicago is an awesome city, and a really cool place to grow up. It'll be an adjustment, but it's worth it! Let me know if you have any more questions, or if you think of any way I can be helpful.