To clone Dolly, researchers took a cell from the udder of a six year old Finn Dorset sheep, and kept in a nutritional solution. They then deprived the cell of the nutrients which made the cell's DNA go into a 'sleeping' state. The nucleus of a sheep egg cell from another ewe was then removed, and inserted the sleeping cell into the now nucleus-free egg cell. The two combined cells with were then given an electric shock, and to the scientist surprise, the combined cells acted like a fertilized egg cell and starting to divideand grow, thus starting the development of Dolly. The egg was then implanted into a third ewe and a few months later, Dolly was born. She was genetically an exact copy of the ewe from which the the udder cell was taken.
There might be a few reasons why Dolly died far too young
Telomere shortening
Telomeres are the ends of a chromosome. As each cell replicates, the telomeres grow progressively shorter. Scientists believe cells stop replicating when the telomeres reach some critical length. Dolly's telomeres were found to be approximately 80% of the length they should be for a sheep her age. It might be for the shortening of telomeres of a cloned animal, you have to count the years of the clone and there mother to get a correct age for it's telomere length. Research is in progress to determine if this phenomenon is seen in other organisms and whether or not it is important.
DNA damaging
Over an organism's lifetime, the DNA within its cells deteriorates, and unrepaired damage causes some diseases, including cancer. Scientists are concerned that this accumulated damage may be carried along during the cloning process. If so, it would give the clone a "head start" on developing some diseases and cancer.