fontsmath-mode
I'm using \usepackage{mathptmx} to use times new roman font in math. But I can't make the font bold by using \boldmath, which works fine with default computer modern font.
I'm using \renewcommand{\rmdefault}{ptm} to use times new roman for the whole document.
How to get bold times new roman font in math so that it blends well with other text?
pslatex is really an obsolete package. You can use mathptmx, as already suggested, or
\usepackage{newtxtext,newtxmath}
(which requires an updated TeX distribution).
mathptmx
newtxtext + newtxmath
Here is the source. Notice how the result is obtained more simply than you did (everybody must learn, when using a new tool).
\documentclass[a4paper]{article}
\usepackage{amsmath} % For maths symbols.
%%% uncomment one of the following and comment the other
\usepackage{newtxtext,newtxmath}
%\usepackage{mathptmx}
\begin{document}
Times New Roman.
$\beta=3\alpha$
$\beta=3\alpha$
$\alpha, \beta, \gamma, \delta, \epsilon, \lambda, \phi, \rho, \varphi, \omega$
\begin{equation}
t_{1/2} = \frac{\ln(2)}{\lambda}.
\end{equation}
\end{document}
(The difference in size is an artifact of the image production.)
TeX Gyre Termes Math has blackboard bold letters similar to the amsfonts ones:
\documentclass{article}
\usepackage{unicode-math}
\setmathfont[range=\mathbb]{TeX Gyre Termes Math}
\begin{document}
$abc\mathbb{R}\mathbb{Q}\mathbb{N}$
\end{document}
Here's the result if XITS Math is used intead of Termes:
Just take your pick; you can experiment also with TeX Gyre Pagella Math or Asana Math.
Best Answer
The mathptmx package is not the best way to load the times font as the fonts are both incomplete and a patchwork of symbols from different sources.
Among the free fonts, you best choice today is probably newtx (alternatively, on older distributions, you have txfonts, but they have some spacing problems):