How to write programming code containing the ‘ character in latex?

Load the upquote package to fix this issue in verbatim mode.

If you want straight quotes in monospaced text mode (e.g., \texttt{...}), or indeed in any other font, then you can use the \textquotesingle command defined in the textcomp package:

\documentclass{article}
\usepackage{upquote,textcomp}
\begin{document}
\newcommand\upquote[1]{\textquotesingle#1\textquotesingle}
\verb|'c'| \texttt{\upquote{h}}
\textsf{\upquote{h}} \upquote{h}
\end{document}

This will work well for fonts in any encoding rather than depending on a specific glyph slot (such as \char13 in the default OT1 encoding).