Does DNA influence death (e.g. by cell aging)
Perhaps the best type of study to examine the role of genes (DNA) in human aging are twin studies.  They have either: the same DNA (monozygotic, one egg, "identical" twins (MZ) ) or similar DNA (dizygotic twins, two eggs (DZ) ) and are perhaps exposed the same, or different environments.   I won't repeat the information in the link, above, but studies of this type guide that our DNA is not an absolute roadmap of our future.

DNA damage can be caused by exposure to excessive radiation, viruses, chemicals and just 'random events'.  Cells and tissues can also be damaged by oxidative damage - literally oxygen and related molecules that are produced within the body and can react with components of cells (e.g DNA, proteins and the lipid cell membrane).   The accumulation of these oxidized products in and around the cell disrupts tissue function and leads to many of the symptoms of aging, a process called sensescence. 

We obviously cannot avoid having oxygen around us.  In the same way, our cells need the sugar glucose as a source of energy. Glucose can spontaneously react with proteins and lipids to form advanced glycation endproducts (AGEs).
 AGEs are pro-inflammatory.  In diabetes, where blood glucose levels may remain at higher than normal for extended periods, the formation of AGEs can lead to nerve and kidney damage.   The chemical reaction that is occurring is the same as when you toast a piece of bread or fry a steak.  So paradoxically, two of the ingredients essential for life - oxygen and glucose - are slowly but surely disrupting our body's function over the course of decades.   Although the body can replace some of these damaged proteins, they accumulate over time, as evidenced by the increasing lack of elasticity of older skin as collagen proteins are cross-linked. 

Regular exercise, good diet and friendships all add to mental and physical health, quality of life and longevity.  So, to live a long time, don't stress, have ONE slice of cake and make new friends who will exercise with you.