Using BibTeX with letter class

Taking the definition from the article class, I get good results with

\begin{filecontents}{letter.bib}
@article{First,
  author  = "Other, A. N.",
  title   = "Some things {I} did",
  journal = "J. Irreproducible Results",
  year    = "2011"
}

@article{Second,
  author  = "Aaa, S{\o}mebloke",
  title   = "Tigers",
  journal = "Ann. Improbable Res.",
  year    = "2011"
}
\end{filecontents}

\documentclass{letter}
\makeatletter
\newenvironment{thebibliography}[1]
     {\list{\@biblabel{\@arabic\c@enumiv}}%
           {\settowidth\labelwidth{\@biblabel{#1}}%
            \leftmargin\labelwidth
            \advance\leftmargin\labelsep
            \usecounter{enumiv}%
            \let\p@enumiv\@empty
            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
      \sloppy
      \clubpenalty4000
      \@clubpenalty \clubpenalty
      \widowpenalty4000%
      \sfcode`\.\@m}
     {\def\@noitemerr
       {\@latex@warning{Empty `thebibliography' environment}}%
      \endlist}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\makeatother
\begin{document}
\begin{letter}{Some person}
\opening{Hello}

Some text \cite{First}, more text \cite{Second}.

\bibliographystyle{unsrt}
\bibliography{letter}

\end{letter}
\end{document}

I've used the unsrt style here, as plain would put the references in alphabetical rather than citation order. I've also modified the definition of thebiliography a little, removing the section-related stuff as this does not really seem relevant to a letter. (I also took out the code related to the openbib option for the article class, again as it does not seem relevant.)

---

The second edit to the question asks about natbib. For me, this works if I load natbib after defining thebiliography and if I make \bibsection 'safe':

\begin{filecontents}{letter.bib}
@article{First,
  author  = "Other, A. N.",
  title   = "Some things {I} did",
  journal = "J. Irreproducible Results",
  year    = "2011"
}

@article{Second,
  author  = "Aaa, S{\o}mebloke",
  title   = "Tigers",
  journal = "Ann. Improbable Res.",
  year    = "2011"
}
\end{filecontents}

\documentclass{letter}
\makeatletter
\newenvironment{thebibliography}[1]
     {\list{\@biblabel{\@arabic\c@enumiv}}%
           {\settowidth\labelwidth{\@biblabel{#1}}%
            \leftmargin\labelwidth
            \advance\leftmargin\labelsep
            \usecounter{enumiv}%
            \let\p@enumiv\@empty
            \renewcommand\theenumiv{\@arabic\c@enumiv}}%
      \sloppy
      \clubpenalty4000
      \@clubpenalty \clubpenalty
      \widowpenalty4000%
      \sfcode`\.\@m}
     {\def\@noitemerr
       {\@latex@warning{Empty `thebibliography' environment}}%
      \endlist}
\newcommand\newblock{\hskip .11em\@plus.33em\@minus.07em}
\makeatother
\usepackage[numbers]{natbib}
\let\bibsection\relax
\begin{document}
\begin{letter}{Some person}
\opening{Hello}

Some text \cite{First}, more text \cite{Second}.

\bibliographystyle{unsrtnat}
\bibliography{letter}

\end{letter}
\end{document}

I had the same problem and I think that I found a more elegant and more extensive solution.

The developers of komascript.de provide a class scrartcl that is basically equivalent to the standard article class but allows the integration of letters. The beauty is, you cannot only print a bibliography but also use sections and other tools that you might have missed from the article class. A working example:

\begin{filecontents}{letter.bib}
@article{First,
  author  = "Other, A. N.",
  title   = "Some things {I} did",
  journal = "J. Irreproducible Results",
  year    = "2011"
}

@article{Second,
  author  = "Aaa, S{\o}mebloke",
  title   = "Tigers",
  journal = "Ann. Improbable Res.",
  year    = "2011"
}
\end{filecontents}

\documentclass{scrartcl}
\usepackage{scrletter}
\usepackage[numbers]{natbib}
\usepackage{lipsum}

\begin{document}
\setkomavar{fromname}{Name of Sender}
\setkomavar{fromaddress}{Address of Sender}
\begin{letter}{Name of Recipient\\Address of Recipient}
  \opening{Dear Recipient,}

  \section{Interesting Stuff}
  The first~\cite{First} citation.
  \lipsum%

  \section{Another Citation}
  And the second~\cite{Second} citation.

  \closing{Yours sincerely,}

  \bibliographystyle{unsrtnat}
  \bibliography{letter}
\end{letter}
\end{document}