Can I count the number of times each reference is cited?

If you're willing to switch to biblatex, you may use the citecounter feature that was added in v1.3.

\documentclass{article}

\usepackage[citecounter=true]{biblatex}

\renewcommand{\finentrypunct}{%
  \addperiod\space
  (Cited \arabic{citecounter}~time\ifnumequal{\value{citecounter}}{1}{}{s})%
}

\usepackage{filecontents}

\begin{filecontents}{\jobname.bib}
@misc{A01,
  author = {Author, A.},
  year = {2001},
  title = {Alpha},
}
@misc{B02,
  author = {Buthor, B.},
  year = {2002},
  title = {Bravo},
}
@misc{C03,
  author = {Cuthor, C.},
  year = {2003},
  title = {Charlie},
}
\end{filecontents}

\addbibresource{\jobname.bib}

\nocite{*}

\begin{document}

Some text \autocite{A01}.

Some text \autocite{B02}.

Some text \autocite{A01}.

\printbibliography

\end{document}

You can use the backref or pagebackref options to hyperref to get links to section or page numbers after each reference in the bibliography if you don't want to switch to biblatex.