Asheville sits in a valley at about 2000' elevation. The BRP runs a little over 11 miles in the area without much elevation change between the French Broad River in the south to US 70 in the north (that would be BRP directions) with mile marker 390 being in this 11 mile run.
To the south the parkway runs 79 miles south with a substantial climb of 2800' in 15 miles to start, running to Mount Pisgah at mile marker 409. Mt. Pisgah is a campground, inn, and restaurant located at the top of the climb. There is tunnels on this climb which will require lights on your bicycle, and I mean lights that will be able to illuminate the road and yourself as the tunnel is long enough to be completely dark. Past Mount Pisgah going south the road rolls and climbs a bit until you reach Richland Balsam at 6053', the highest point on the parkway. From there it descends quite a bit before you have a substantial climb to Waterrock knob. The last bit is all downhill to Cherokee which is also about 2000' in elevation. There are a few tunnels on the descent into Cherokee.
Going north from Asheville there is a long climb to Craggy Gardens of 19 miles and 3400' feet. Beyond Craggy Gardens it rolls a bit before descending down to NC 80. The turn off to Mount Mitchell is in this section, a five mile side road that will take you to the highest point in the US east of the Mississippi. There is a state park and small campground there. On the parkway after NC 80 there is some climbing that takes you to Crabtree Meadows, but the campground there has not been open the last couple of seasons. A few miles beyond that is Little Switzerland which has motels, inns, and restaurants. Little Switzerland is at mile marker 335. The tunnels in this direction are not as dark and blinky lights will probably suffice.
Which direction is best? I wouldn't try to choose that, but they are both character building. It just depends on what you have in mind, which is hard to tell from your short question. If you are looking to do a lot of riding on the parkway there is more going north as it is possible to ride the whole distance southward to Cherokee in a day. If you are looking for an overnight then a night at Mount Pisgah is nice, either camping or in the inn. Going north you can always do the climb to Mount Mitchell and then tell your friends you have ridden to the highest point in the east. Either directions is remote and very few services. I have only done the parkway in this area northbound, so I can't offer a direct comparison from experience.
