Without changing the code:
Add an extra row at the bottom, merge the cells with \multicolumns and put the footnote manually there, i.e. format it with superscript and footnote text. A MWE (NB! Ugly spacing):
\documentclass{article}
\begin{document}
\begin{table}[h]
\centering
\begin{tabular}{lccc} % creating 4 columns
\hline \\
test & test & test & test\textsuperscript{*} \\ % here is the info
\hline
\multicolumn{4}{l}{\textsuperscript{*}\footnotesize{The footnote}}
\end{tabular}
\caption{\textbf{Performance at First Doubles Matches}} % title name of the table
\label{tab:PPer}
\end{table}
\end{document}
NB!!: Please, use the package booktabs to get nicer spacing and lines.

A prettified MWE_
\documentclass{article}
\usepackage{booktabs}
\begin{document}
\begin{table}[h]
\centering
\begin{tabular}{@{}lccc@{}} % creating 4 columns
\toprule
\textbf{H1} &\textbf{H2} & \textbf{H3} & \textbf{H4} \\ \midrule
testtest & test & testtest & testtesttesttest\textsuperscript{*} \\ % here is the info
testtest & test & testtest & testtesttesttest\textsuperscript{**} \\
testtest & test & testtest\textsuperscript{***} & testtesttesttest \\\midrule
testtest & test & testtest & testtesttesttest\\
testtest & test & testtest & testtesttesttest\\ \midrule[.5pt]
\multicolumn{4}{l}{\textsuperscript{*}\footnotesize{The first footnote}}\\
\multicolumn{4}{l}{\textsuperscript{**}\footnotesize{The second footnote}}\\
\multicolumn{4}{l}{\textsuperscript{**}\footnotesize{The third footnote}}
\end{tabular}
\caption{\textbf{Performance at First Doubles Matches}} % title name of the table
\label{tab:PPer}
\end{table}
\end{document}

Changing the code
Use the package threeparttable, or use the package longtable, which support footnotes and can be used even if the table does not span pages. You have to change you code, though.