How can I put a period right after the algorithm's number in the algorithm's title?

Use caption to change the labelsep method to period. Since the original caption sets the algorithm label in bold, we also change labelfont to bf.

\documentclass{article}

\usepackage{algorithm,caption}
\usepackage{algpseudocode}

\captionsetup[algorithm]{
  labelfont = bf,
  labelsep = period
}

\begin{document}

\begin{algorithm}
  \caption{Some random algorithm}
  \begin{algorithmic}[0]
    \Procedure{random}{}
    \EndProcedure
  \end{algorithmic}
\end{algorithm}

\end{document}

The usual way seems to work. (See here for further examples.)

\documentclass{article}

\usepackage{algorithm}
\usepackage{algpseudocode}

\begin{document}
\renewcommand\thealgorithm{\arabic{algorithm}.}
\begin{algorithm}
\caption{Some random algorithm}
\begin{algorithmic}[0]
\Procedure{random}{}:
\EndProcedure
\end{algorithmic}
\end{algorithm}
\end{document}