To remove the vertical bar, add \multicolumn{1}{c|}{} to the first cell. The reason for the different column widths is described in Table column widths disproportionate due to multicolumn cell being too long A workaround is shown below.

\documentclass{scrartcl}
\usepackage{array}
\begin{document}
\begin{tabular}{|c| *{4}{>{\centering\arraybackslash}p{0.7cm}|}}
\cline{2-5}
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{Method 1} & \multicolumn{2}{c|}{Method 2} \\
\hline
N & E & SE & E & SE \\
\hline
10 & 3 & 1 & 0 & 1 \\
100 & 2 & 0 & 0 & 1 \\
1000 & 3 & 1 & 1 & 1\\
10000 & 3 & 1 & 1 & 1 \\
\hline
\end{tabular}
\end{document}
For good measure, here's a different way of laying out the table, without using vertical rules. \toprule, \cmidrule, \midrule and \bottomrule are provided by booktabs.

\documentclass{scrartcl}
\usepackage{array,booktabs}
\begin{document}
\begin{tabular}{c *{4}{>{\centering\arraybackslash}p{0.7cm}}}
\toprule
\multicolumn{1}{c}{} & \multicolumn{2}{c}{Method 1} & \multicolumn{2}{c}{Method 2} \\
\cmidrule(rl){2-3} \cmidrule(rl){4-5}
N & E & SE & E & SE \\
\hline
10 & 3 & 1 & 0 & 1 \\
100 & 2 & 0 & 0 & 1 \\
1000 & 3 & 1 & 1 & 1\\
10000 & 3 & 1 & 1 & 1 \\
\bottomrule
\end{tabular}
\end{document}