I'm no expert on orbital mechanics, but the book went somehing like this.
Find your chosen asteroid (carbonacious condrite?) then travel to it conventionally (ie spacecraft)
Plant a few rockets on the thing and fire them off to change orbit to put the asteroid into Geo orbit (or close to it, safer to aim high and refine it once it has arrived on site) This would take a few years in order to avoid a huge fuel cost.
Once your asteroid is in geo orbit, you set up a manufacturing plant on it, meanwhile ship up 25000-30000 miles of magic super wonder cable attach one end to asteroid, and lob the end at the Earth. (what you lob down is a re-entry capsule shaped a bit like apollo ones, but no-one inside, did have parachutes in the book though, to avoid very high touchdown speeds.
Anchor Earth end of cable, then you build the actual elevator system down from the asteroid, using the manufacturing plant that you've built on the asteroid.
Raw material for wonder cable is carbon, which is what the asteroid is made from, so supplies to the space station/asteroid are limited to people, machinary, and air/food/water. Basically until system is up and running, most supplies come from conventioal rockets.
Supercarb - I shall have to remonstrate with my old school librarian, doubt she'll be interested 15 years after I left though
