Well, seeing that the word "deathly" features prominently in our title, you can kind of figure out that death and mortality would play a pretty big role in this book. Basically, Harry is forced to finally confront the specter that's been haunting him all these years – death – face-to-face. While death has always shadowed his life since the murder of his parents, now it's right up close and (all too) personal. From the deaths of many beloved friends to the requirement that Harry himself shuffle off this mortal coil, he (and we, his readers) are constantly forced to try and understand what death is, what is means, and how to deal with it.
Death has always played an important role in the Harry Potter books, a fact that Rowling attributes, in part, to her mother's death while she was working on her first book:
Definitely Mom dying had a profound influence on the books because […] in the first draft, [Harry's] parents were disposed really in quite […] in the most cavalier fashion. I didn't really dwell on it. Six months in my mother died and I simply [couldn't kill off the fictional] mother. That callously. Not – it wasn't callous, but it's – it wasn't what it became [...] And I really think from that moment on, death became a central, if not the central, theme of the seven books. […] The theme of how we react to death, how much we fear it. Of course, I think which is a key part of the book because Voldemort is someone who will do anything not to die. He's terrified of death. And in many ways, all of my characters are defined by their attitude to death and the possibility of death. (source)