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.