What is a Visual Climb Over Airport (VCOA) and how is it properly executed?
It's defined in the pilot/controller glossary:


  VISUAL CLIMB OVER AIRPORT (VCOA)− A departure option for an IFR
  aircraft, operating in visual meteorological conditions equal to or
  greater than the specified visibility and ceiling, to visually conduct
  climbing turns over the airport to the published “climb−to” altitude
  from which to proceed with the instrument portion of the departure.
  VCOA procedures are developed to avoid obstacles greater than 3
  statute miles from the departure end of the runway as an alternative
  to complying with climb gradients greater than 200 feet per nautical
  mile. These procedures are published in the ‘Take−Off Minimums and
  (Obstacle) Departure Procedures’ section of the Terminal Procedures
  Publications.


The AIM 5-2-8 says that pilots must request it:


  Pilots are responsible to advise ATC as early as possible of the
  intent to fly the VCOA option prior to departure


And the ATC orders 4-3-2 say that ATC may not offer it:


  Do not solicit use of the Visual Climb over Airport (VCOA) option.


The TERPS 14-4 has all the technical details and explains the purpose, which is to allow aircraft that can't maintain a 200fpnm climb to climb safely within a limited area over the airport:


  VCOA is an alternative method for pilots to depart the airport where
  aircraft performance does not meet the specified climb gradient.


The departure designer creates a vertical cylinder (called a Visual Climb Area in the TERPS) of airspace over the airport that's safe to climb in, based on airport elevation, runway positions and a default airspeed of 250kt (although other airspeeds can be used). Even if the VCA dimensions aren't charted or stated in the DP, for an airport at 2000msl (KAVL is at 2162msl) the radius of the cylinder is at least 2nm at 90kts, or 2.8nm at 250kts, which seems to be plenty for a spiral climb (in the aircraft I fly, anyway). And the climb has to be in visual conditions, so terrain clearance shouldn't be an issue.

I don't know of any detailed guidance on how execute it, but "climbing turns over the airport" seems fairly clear. I've never done it myself, but I'd simply fly a continuous spiral climb after takeoff, up to the en route altitude mentioned in the procedures and go from there.