I'd look just inside or just outside the Beltline on the the NW arc of the city between Lake Boone Trail and Wake Forest Rds. Depending on your budget and the sort of neighborhood you desire, you might want to check out the following:
- North Hills
- Drewey Hills
- Country Club Hills
- Brookhaven
- Quail Hollow
- Anderson Heights
Closer to Downtown than the above, you might also want to look into Hayes Barton, Five Points and Cameron Village and Cameron Park (my favorite).
All of the above are well within the city limits of Raleigh and are established neighborhoods.