pgoat:
You'll be happy with bike commuting in Chapel Hill.
(though you may have to deal with
this monstrosity).
The worst areas for bike commuting might be 15-501 and 54 west (connecting west Durham with Chapel Hill and Carrboro), 54 east (connecting Chapel Hill to south Durham), and MLK Blvd (north, toward Hillsborough). Those are major arteries; some of them have secondary roads paralleling them. Franklin Street just north of downtown can be tough too; it's hilly and twisty so it's hard for cars to see you. Manning Drive, in front of UNC Hospitals, gets lots of motorist traffic.
Try Manning Drive at rush hour or around the time of hospital shift changes (7 AM, 7 PM), or just before or after some big event at the Dean Smith Center.
Generally, Chapel Hill is a bike-friendly mid-sized town. Being thoroughly dominated by a large university helps in that regard. In the general area, Carrboro is similarly bike-friendly (if not more so); central Durham is very bike-friendly (once you get away from the big-box stores); Raleigh and Cary seem somewhat less so.