When UV light falls on the skin it has to go through the cell membrane and the nuclear membrane to reach the chromosomes. So it looks like that the DNA is protected but it probably isn't. Won't membranes be damaged first and the so that the cell dies before the DNA is affected and is replicating wrongly (if it is damaged too)? Or is DNA just really much more vulnerable than cell membrane?
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).
When UV light is on your skin it has to go through the cell membrane and to the membrane of the core. So it looks like that the DNA is protected.
I don't understand what you mean by membrane of the core (perhaps nuclear membrane) but yes there is cell membrane and other cytosolic components. You should, however note that not all cells are spherical; some cells like skin cells are flat and along the thickness axis the DNA is much "less buried".
However, UV light does get attenuated by different cellular components. Melanin (which gives the skin a tan), for instance, absorbs UV and protects the DNA.
In spite of all this UV can reach the DNA. Furthermore, DNA (basically the bases) can absorb UV because of the nature of their chemical structure. After absorbing UV they reach an excited state and can become reactive. Reactions such as pyrimidine dimerization happens because of UV absorption. Note that this kind of an effect is elicited only by high energy UV (200–300 nm) (Durbeej and Eriksson, 2002; Svobodová et al., 2012).
Low energy UV (UVA) does not damage the DNA directly but can do that indirectly by generating free radicals. These free radicals are generated when other cellular components absorb UV and become reactive. Lipids are also affected by UVA and UVB; lipid peroxidation happens in the presence of UV and these peroxides are a source of free radicals (Morliere et al., 1995). So, the membrane is indeed "damaged".
It is also interesting to note that UVA is the most available to the skin cells because UVC and UVB are filtered to a great extent by the ozone layer and also scattered/absorbed by skin components (What is the relationship between UV wavelength and penetration depth into human skin?).
To summarize, DNA is more vulnerable to certain UV ranges but the damage frequently happens indirectly by other free radicals generated in the cytosol.