Do people who use the Death Note become Shinigami themselves?
No.

Humans who die don't go to Heaven or Hell (because neither exists).

They go to Mu (Which means, nothing).


  Death Note, Volume 12, page 188, has these two rules in regards to
  "Mu": "All humans will, without exception, eventually die." Also,
  "After they die, the place they go is MU (Nothingness)."




Basically, there's nothing after death. Shinigami are a different species.