Close.

    if($count < 5)
    {
        ...
    }
    else
    {
        print "$pos ... \n";
        $count = 0;
        @data  = ();
    }
[download]

but you would need to reset $largest_cons<n> too, depending on your requirements.

Some points:

• you use English, FileHandle and Exception, but then you don't make use of them. Why?
• Why those $largest_cons<number> variables? Wherever you are inclined to number your variables, you really want an array
• at open(FILE, "$input") use three-argument open
• at open(FILE, "$input") - useless use of quotes

You could also use $. (see perlvar) in a flip-flop (".." - see perlop):

#!/usr/bin/perl -w

use strict;
use warnings;

my $input = shift;

#open FILE, '<', "$input" or die "ERROR: Unable to open input file: $!
+\n";

my @largest_cons = (0) x 6; # inhibit "uninitialized" warnings

while (<DATA>)
{
    my @data = split;

#    if ( 1 .. 5 )    # see update below
#    {
        $largest_cons[0] = $data[0]
          if $. == 1;
        for (1..$#data)
        {
            $largest_cons[$_] = $data[$_]
              if $data[$_] > $largest_cons[$_];
        }
        if ($. == 5) 
        {
            $largest_cons[0] .= '-' . $data[0];
            print "@largest_cons\n";
            $. = 0;
            @largest_cons = (0) x 6;
        }
#    }
}
[download]

Note that starting with 1, you end at 1-5 .. 11-15 rather than 10-14. For that you need a row 0.

Update: on a second look at the code I've posted, the flip-flop-business doesn't make sense here... ;)