Here's one option using the pdfpages and background packages; your original.tex file looks something like
\documentclass{letter}
\usepackage[a6paper]{geometry}% just for the example
\usepackage{pdfpages}
\usepackage[some,contents=Copy]{background}
\usepackage{lipsum}
\signature{Your name}
\address{Street \\ City \\ Country}
\pagestyle{empty}
\begin{document}
\begin{letter}{Company name \\ Street\\ City\\ Country}
\opening{Dear Sir or Madam:}
\lipsum[4]\lipsum[4]
\closing{Yours Faithfully,}
\ps{P.S. Here goes your ps.}
\encl{Enclosures.}
\end{letter}
\includepdf[pages={-},pagecommand={\BgThispage\pagestyle{empty}},noautoscale]{copy.pdf}
\end{document}
I used the letter document class, but you can use any other of the more versatile classes around.
You first process the document commenting out the line:
\includepdf[pages={-},pagecommand={\BgThispage\pagestyle{empty}},noautoscale]{copy.pdf}
Then you copy the resulting PDF file to copy.pdf using (on Linux):
cp original.pdf copy.pdf
Then you uncomment out the line
\includepdf[pages={-},pagecommand={\BgThispage\pagestyle{empty}},noautoscale]{copy.pdf}
And reprocess original.tex.
An image of the result for the above example:
