Dying every time you go to sleep is a wild exaggeration of the experience of consciousness primarily because your brain keeps working, it never actually 'stops' it just slows down. And the step of killing the original you isn't necessary in any way if you assume you've made a perfect copy with a good 'push' to get it going. Unless you think the soul is required, then a clone would just be 'another' you whether the original is dead or not. It would be interesting to see just how quickly the two copies deviated from each other, but more from a scientific standpoint of brain plasticity rather than an exercise in 'who it really who'. Even tiny differences in experience see likely to start deviating the continuing consciousness very quickly.