Yes, you have to calculate the total energy, but it's still just a few dollars per pound.
How do you lift the string and counterweight in the first place? Why would you even need a string and counter weight?
You lift it with rockets, though the counterweight could be an asteroid that you corral, or perhaps a lump of lunar material.
You need the string to provide a basic structure for the elevator, just as you need to throw a line over a gorge to start building a suspension bridge. The counterweight is to keep the system in tension. Objects below geosynchronous altitude will have positive weight (they'll want to fall down) and objects above will have negative weight (they'll want to "fall" up).