How to implement continuations?

A good summary is available in Implementation Strategies for First-Class Continuations, an article by Clinger, Hartheimer, and Ost. I recommend looking at Chez Scheme's implementation in particular.

Stack copying isn't that complex and there are a number of well-understood techniques available to improve performance. Using heap-allocated frames is also fairly simple, but you make a tradeoff of creating overhead for "normal" situation where you aren't using explicit continuations.

If you convert input code to continuation passing style (CPS) then you can get away with eliminating the stack altogether. However, while CPS is elegant it adds another processing step in the front end and requires additional optimization to overcome certain performance implications.

I remember reading an article that may be of help to you: Cheney on the M.T.A. :-)

Some implementations of Scheme I know of, such as SISC, allocate their call frames on the heap.

@ollie: You don't need to do the hoisting if all your call frames are on the heap. There's a tradeoff in performance, of course: the time to hoist, versus the overhead required to allocate all frames on the heap. Maybe it should be a tunable runtime parameter in the interpreter. :-P

If you are starting from scratch, you really should look in to Continuation Passing Style (CPS) transformation.

Good sources include "LISP in small pieces" and Marc Feeley's Scheme in 90 minutes presentation.