Lifespan is a function of organism size (ironically measured in weight, mostly). The bigger an organism, the longer it's lifespan. Keeping in mind normal human lifespan would be about 30-35 years, you find that many animals of similar size have a natural lifespan within a factor 2 or so of a formula that also applies to humans. Normal whales, for instance, have a natural lifespan of about 70 years.

That this formula works would seem to indicate that you are right. Death is natural, but it's "planned". The easiest way for such a formula to work would be that your genes somehow contain a death clock.

But there are multiple death clocks. One limits number of cell divisions. There is another one known that limits the amount of energy that can pass through a cell, after which it will kill itself. There are various others, one that kills the cell if it isn't deactivated on a regular basis (presumably meant as a check on DNA integrity), one that is triggered from the outside of the cell, ... the list goes on.