How can I get bold Greek letters with mathptmx?

As a last resort, over-printing (or under-printing) as a method to simulate boldness (or fake bold) is an option using the contour package. It may come in handy when there is no bold version of a font/symbol available:

\documentclass{article}
\usepackage{mathptmx}% http://ctan.org/pkg/mathptmx
\usepackage{array,contour}% http://ctan.org/pkg/{array,contour}
\begin{document}
\begin{tabular}{rl@{\quad}>{\contourlength{0.01em}}l}
\verb|\sigma|: & $\sigma$ & $\sigma$ \\
\verb|\contour[1]{red}{$\sigma$}|: & \contour[1]{red}{$\sigma$} & \contour[1]{red}{$\sigma$} \\
\verb|\contour[10]{black}{$\sigma$}|: & \contour[10]{black}{$\sigma$} & \contour[10]{black}{$\sigma$} \\
\verb|\contour{black}{$\sigma$}|: & \contour{black}{$\sigma$} & \contour{black}{$\sigma$} \\
\verb|\contour*{black}{$\sigma$}|: & \contour*{black}{$\sigma$} & \contour*{black}{$\sigma$}
\end{tabular}
\end{document}

The default interface is \contour[<number>]{<color>}{<stuff>} that prints <number> copies of <stuff> under <stuff> using colour <color>. The default is 16 repetitions if none are specified, while the starred-version \contour* prints 32. \contourlength{<len>} sets the radius of the under-printing. The two column above are default (0.03em) and 0.01em.

To make the under-printing more obvious/clear, here's a close-up view of the above contours at the default setting:

And, for \contourlength{0.01em}:

With newtx you can get bold math symbols; the text font is very similar to what you'd get with mathptmx and the math fonts are surely much better.

\documentclass{article}
\usepackage{newtxtext,newtxmath}
\usepackage{bm}

\begin{document}
$\Sigma\sigma$

$\bm{\Sigma}\bm{\sigma}$

$\sum\bm{\sum}\displaystyle\sum\bm{\sum}$
\end{document}

Note that \boldsymbol provided by the amsbsy package is to be considered obsolete as \bm is much better.