Change the definition of the macros \acfa and \@acf (EDIT: and their plural couterparts).
\documentclass{article}
\usepackage{acronym}
\makeatletter
\renewcommand*{\acfa}[1]{%
% \texorpdfstring{\protect\@acf{#1}}{\AC@acl{#1} (#1)}}% DELETED
\texorpdfstring{\protect\@acf{#1}}{#1\ (\AC@acl{#1})}}% NEW
\renewcommand*{\@acf}[1]{%
\ifAC@footnote
\acsfont{\AC@acs{#1}}%
\footnote{\AC@placelabel{#1}\AC@acl{#1}{}}%
\else
\acffont{%
% \AC@placelabel{#1}\AC@acl{#1}% DELETED
\AC@placelabel{#1}\AC@acs{#1}% NEW
\nolinebreak[3] %
% \acfsfont{(\acsfont{\AC@acs{#1}})}% DELETED
\acfsfont{(\acsfont{\AC@acl{#1}})}% NEW
}%
\fi
\ifAC@starred\else\AC@logged{#1}\fi}
\renewcommand*{\acfpa}[1]{%
\texorpdfstring{\protect\@acfp{#1}}{\AC@acsp{#1} (\AC@aclp{#1})}}
\renewcommand*{\@acfp}[1]{%
\ifAC@footnote
\acsfont{\AC@acsp{#1}}%
\footnote{\AC@placelabel{#1}\AC@aclp{#1}{}}%
\else
\acffont{%
\AC@placelabel{#1}\AC@acsp{#1}%
\nolinebreak[3] %
\acfsfont{(\acsfont{\AC@aclp{#1}})}%
}%
\fi
\ifAC@starred\else\AC@logged{#1}\fi}
\makeatother
\begin{document}
Some text about the \ac{PC}~\dots
More about the \ac{PC}~\dots
\begin{acronym}
\acro{PC}{Personal Computer}
\end{acronym}
\end{document}