If your document doesn't load the hyperref package, then adopting the recommendation in @PeterWilson's answer -- to load the footmisc package with the option multiple -- is all you need to do.
If, however, your document does load the hyperref package, the aforementioned approach no longer works, i.e., the commas between adjacent footnotes disappear mysteriously. A different approach is required. Specifically, the footnote macro needs to be modified along the lines proposed in this answer to the posting Incompatibility between footmisc-option multiple and hyperref.
As an alternative to modifying the \footnote macro yourself (as is done in the code below), just run
\usepackage[dont-mess-around]{fnpct}
in the preamble.

\documentclass{article}
\setlength\textheight{3cm} % just for this example
\let\oldFootnote\footnote
\newcommand\nextToken\relax
\renewcommand\footnote[1]{%
\oldFootnote{#1}\futurelet\nextToken\isFootnote}
\newcommand\isFootnote{%
\ifx\footnote\nextToken\textsuperscript{,}\fi}
%\usepackage[multiple]{footmisc} % doesn't hurt, but doesn't work either
\usepackage[colorlinks,allcolors=blue]{hyperref}
\begin{document}
Some text.\footnote{First}\footnote{Second}\footnote{Third}\footnote{Fourth}\footnote{Fifth}
Some more text.
\end{document}