Consciousness is a mechanism of the brain and, scientifically speaking, there is no mystical 'soul' that holds ones identity. If you were to replicate a body it would be the same as replicating a chair. They might both look the extremely similar (if you look closely enough there will always be differences) but they are two separate objects.
As far as the first-person experience would go, if both the person (A) and the close (B) woke up in identical beds in identical rooms, they would both have memories of going into whatever procedure it was that manufactures the clone and wake up believing they were A. B wouldn't know the difference until the differences were revealed to them. Also, A would not experience conscious over two bodies. A would have no added experience, beyond whatever occurred during the procedure. To A, B would just be a different person that seems extremely similar to them. B would feel the same way. Credible evidence would have to be procured to prove who is A and who is B. From then on, their identities become the result of whatever experiences they accumulate and they would diverge into difference.
However, it gets tricky.
Just say you saved an instance of all possible information contained within your body and saved it into some computer, ready to clone again at some point in the future. Every clone thence created would wake up believing that they had just finished the procedure to save the information of their body, and they would all believe "I'm me" until credible evidence were provided to prove that they are in fact a clone.
Consciousness is a continuity backed up by continually degrading memories.
A neat podcast on this:
http://philosophybites.com/2008/11/christopher-shi.html (or right click and save as ->
http://cdn1.libsyn.com/philosophybites/Christopher_Shields_on_Personal_Identity.mp3?nvb=20081103070152&nva=20081104070152&t=0835855fc55c33a31923f )
