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I'm moving in from out of state, and I will not be able to visit to tour any apartments or anything. The area is really nice overall compared to where I live, so I'm not too concerned about safety or cleanliness, and I've already lived in some apartments with crappy landlords, dealing with maintenance, etc. You can also have bad neighbors no matter where you go. My plan is to find a nice-seeming place that is conveniently located and sign a year lease. If I don't like the apartment I can spend a lot of time on campus and move to a new apartment next summer.
To find a place, I've looked at google maps and typed the name of the building where I'll have my classes, then "apartments." Carrboro area is where a lot of grad students live. Durham is good for grad students too because it's inexpensive due to a bad reputation in the past, but over the last decade or two things have gotten really nice over there. This is information I got from students and professors. 
You can also search on apartments.com or rent.com, but once you find a place you would want to go to the apartment's website rather than dealing with apartments.com or rent.com. I'd always rather deal with the apartment management than a third party website. 
One thing I've noticed that I have never seen before is that rent goes up and up and up every day/week/month closer we get to August. They know students will need to find a place ad take advantage of the desperation we have for housing and jack up the prices. It's a good idea to try to secure a place ASAP rather than wait. 

Nice to electronically meet you all!  I'll be moving to Chapel Hill mid-May right after graduation.  I'm originally from Chapel Hill, and my parents still live there, so I'll be moving home until I can find an apartment.  If anyone will be in the area over the summer, let me know, because I'd love to meet some people before classes start.

I'm planning to get to Chapel Hill by the first/second week of August - there are some mandatory events to attend for international students (mostly to do with visa paperwork) around the 16th. Got my I-20 a few days ago, all I need to do now is wait until early/mid May to file in the application with the Consulate in Sydney and get the visa issued. I'm a big NBA fan, so I'm pretty excited to go to some games at the Dean Dome (I managed to watch the NCAA title game against Gonzaga last week - ugly game, but a victory nonetheless)! 

Hello from Sydney, Australia! Very excited to head over in autumn (I know you Americans refer to it as "fall"...). Won't disclose publicly which program to which I've been admitted since it's a small department and there's a good chance I will be identified immediately, but I'll be sticking around for the next 4-5 years to get a PhD.