Sure, pondering what will happen to your physical form after you're gone is pretty morbid. But, it's a natural thing to be curious about — especially during this here spooky season. And, thankfully, the American Chemical Society's Reactions channel is willing to delve deep into the science-y side of death (with suitably eerie music).
First, the obvious: When you die, your heart and lungs stop working. This causes your blood to thicken and your lungs to stop taking in oxygen. Without circulation, gravity causes your blood to settle and discolor the skin in a process called livor mortis. The lack of moving blood also causes your body temperature to drop and become rigid (a.k.a. rigor mortis).
Without oxygen, your cells can't create their energy source (ATP) and they also die. Then, they begin to break down and release lysosomal enzymes. These make your decomposing body a lovely home for fungi and bacteria, which then cause the putrefaction process to begin. Those bacteria release the chemicals putrescine and cadaverine, which both help to create that patented dead-body smell. Unless the body is otherwise destroyed (e.g., cremated) the decomposition process will continue from there. But, no, your hair and nails won't keep growing.
So, dying (and what follows) certainly sounds unpleasant. But, hey — at least you won't be around to deal with it.