Here is a decent link to an article by Simon Thompson who has already been mentioned (who ironically is not an area native but is from the UK!):
http://www.romanticasheville.com/birds.htm#.UXuqPbWG1rM
In addition to what's been said already, I would also tell you that
The Blue Ridge Parkway runs through Asheville, and is VERY HANDY as a quick route to a fast getaway from the city area of Asheville.
Not specific to birding (but mentioned prominently in the NC Birding Trail book/website) but this is a "must see" part of the Asheville area for out-of-town folks, in my humble opinion... Also it is some of the finest hiking in NC.
Within 60 minutes of Asheville going NORTH is:
*Mt. Mitchell (the highest mountain east of the Mississippi River in the US) which is VERY ACCESSIBLE from a parking lot and walking trail
*Craggy Gardens (a very "alpine" looking area of NC and you should see plenty of Rhododendron in bloom or close to bloom then)
Within 2 hours driving time (but plan longer-as much as double or triple that-as you will want to stop many times along the way at overlooks and attractions) is Linville Falls and Linville Gorge a VERY rugged area for hiking but very beautiful and unique as to biodiversity.
Within 60 minutes of Asheville going SOUTH is Mt. Pisgah, and the Shining Rock and Middle Prong Wilderness areas, which include The Graveyard Fields, a BEAUTIFUL (IMHO) area. This is a GREAT hiking area where you can just stretch your legs for 5-10 minutes, or do a 5-6 hour hike to Cold Mountain... This area looks more like Canada or the American West than NC, and in fact is considered the southernmost reaches of the Canadian Vegetation Zone.
The terrain here is very easy (in MOST places) and I hike here often (even at 375 pounds I can easily do some of the trails...)
NOTE: This is a high elevation area (for the Eastern US anyway) and weather can change fast. Bring an extra layer of clothing, a rain jacket (even if you think you don't need it!) and a daypack with water and snacks. Also, bear in mind that many facilities along the parkway are closed due to budgetary reasons. You can hop off the Parkway in some places for gas or food, but these intersecting roads may be few and far between so make sure you have a full tank of gas and some snacks in case you get sidetracked with the area's beauty.
Let me know if you need or want more info on any of these suggestions...