fancyhdr works with resume (which is based on article), but some adjustments are necessary:
\documentclass[margin,line]{resume}
\usepackage{kantlipsum} % to provide mock text
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\makeatletter
\fancyhead[L]{\textbf{\expandafter\@gobble\@name}}
\makeatother
\fancyhead[R]{\thepage}
\fancyheadoffset[L]{\sectionwidth}
\setlength\headheight{12pt}
\setlength\headsep{3pt}
\addtolength\topmargin{-15pt}
\AtBeginDocument{\thispagestyle{empty}}
\begin{document}
\name{\Large My name}
\begin{resume}
\kant
\end{resume}
\end{document}
Comment out the \fancyheadoffset line if you don't want that the line on the pages following page 1 are extended on the left. If you want to increase the separation between the rule and the text, modify the value of \headsep and also the value of \topmargin.

Note. The mysterious \expandafter\@gobble\@name is needed because \@name expands to \Large My Name, after \name{My Name}, so we have to remove the unwanted \Large command.
fancyhdrpackage. fancyhdr package – azetina Sep 20 '12 at 15:38