[DllImport( "user32.dll" )]
private static extern IntPtr GetWindowDC( IntPtr handle );
[DllImport( "user32.dll" )]
private static extern IntPtr ReleaseDC( IntPtr hWnd, IntPtr hDC );

private void button1_Click( object sender, EventArgs e )
{
    IntPtr hWnd = IEのウインドウハンドル
    IntPtr hDC = GetWindowDC( hWnd );// またはGetDC
    Graphics g = Graphics.FromHdc( hDC );

    //好きに描画
    g.FillRectangle( SystemBrushes.ControlText, new Rectangle( 0, 0, 100, 100 ) );

    g.Dispose();
    ReleaseDC( hWnd, hDC );
}