Which philosophers have influenced Death Note? Which philosophers have influenced Death Note? What is the name (or category) of their philosophies? I want to read deeper.
It doesn't seem to have been spun from an actual person in history or a piece of writing. From an interview with the creator of Death Note Tsugumi Ohba he says this:


  There was nothing in particular. I started thinking of some ideas and
  while these ideas were still floating in the back of my mind, I got
  more ideas to fill up detailed plots like the rules and the god of
  death and so on. Eventually, I thought there was enough material for a
  good story to tell.


Since Death Note is centered around Gods of Death or Shinigami and since there are many urban legends on Shinigami in Japan and lets not mention that Shinigamis are very commonly known about in Japan, it is safe to assume that these may have influenced or helped influence the creation of Death Note, although if there were any actual people in history or a piece of writing that influenced Death Note, then these origins are yet unknown.

and also


  No, there was nothing that I strongly wished to express.
  The basic underlying idea was that “Humans are not immortals and once they are dead, they do not come back alive again”. This is to indirectly say that we should all treasure the present and live our lives to the fullest.


From the text above, this gives some more of an insight as to what also helped influence Death Note (which would have been his own personal thoughts). Refer to sentence highlighted in bold.

In conclusion Death Note was not inspired by any philosophers (people or a writing), since in the comment that was what you were looking for.

Bibliography for reference


http://www.gaiaonline.com/guilds/viewtopic.php?t=21396363


This should help if you want to review my answer, compare it with other sources or to verify that information is correct.