I'm not entirely sure what you're trying to do. Surely, all lines in your file will be between two lines beginning with h, aside from those before the first line beginning with h, and after the last line beginning with h?

It might make it easier to understand the problem if you give some sample data, and the result you expect to get. Have a look through this: How do I post a question effectively?

Finally, based on your question, my best guess would be something like this:

#!/usr/bin/perl 
use strict;
use warnings;

my $file = $ARGV[0];
my $check = 0;
my @line;

while (<DATA>)
    {
        @line = split('', $_);
        $check += 1 if $line[0] eq 'h';

        if ($check%2 == 1)
            {
                if (($line[0] eq 'j') || ($line[0] eq 'E') || ($line[0
+] eq 'G'))
                    {
                        print STDOUT "$_";
                    }
            }
    }

__DATA__

aaaaa 1
bbbbb 2
hhhhh 3
fffff 4
rrrrr 5
lllll 6
jjjjj 7
HHHHH 8
EEEEE 9
GGGGG 10
hhhhh 11
jjjjj 12
EEEEE 13
GGGGG 14
[download]

I hope that helps.