How does UV light damage the DNA when the chromosomes are deep inside the cell?
UV Rays kill the cells by damaging the DNA. UV lights do not disrupt the cell membrane. If a cell is exposed to UV light, it creates THYMINE dimers (bond). Thymine dimers are the actual disruption in the kinks of DNA. UV exposure to skin is proportional to the cell damage. 

P53 is a gene product which takes care of fixing cell damage. However it has a tolerance. If the damage is less, P53 sends damage repair machinery. If it's non-fixable, then P53 directs the cell to programmed death.

More UV exposure, more thymine dimers causing more cells to die (cell death). If the damage is not extensive, then that may cause cancerous cells (Result of cell damage).