First step is to use an airship to launch a payload vertically to an enormous airship (think two mile wide starfish with thin arms), the Dark Sky Station (DSS), that serves as a sob-orbital station at 140,000 ft altitude. The payload is then transfered to a third airship, the V-shaped Orbital Ascender, that will accelerate to insertion velocity and put the payload into orbit.
They have actually already experimented with some of the more simple technology they need by ascending a smaller structure (70 ft across) to altitude multiple times.
An old, old animation:
A PDF explaining it in more detail: ATO -- Airship To Orbit