You need to make your tags before you do your insert, then specify the tag for each line as you insert it.

# while (<FH>) { $right_frame->insert("end", $_); }
 # should be something like this
while (<FH>) { $right_frame->insert("end", $_, $tag); }
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Here is a complete simple example.

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

open (FH,"<$0") or warn "$!\n";
my @lines = <FH>;
close FH;

my $mw  = MainWindow->new(-title=>'Colored Text');
$mw->geometry(($mw->screenwidth-10) . 'x' .
             ($mw->screenheight-10) . '+0+20');

$mw->fontCreate('big',
   -family=>'arial',
  -weight=>'bold',
  -size => 25 );

my $text = $mw->Scrolled("Text",
          -scrollbars => 'osw',
          -background => 'white',
          -font =>'big',
      -wrap => 'none',
                          )->pack;

$text->tagConfigure('greyline', -background => 'grey95');

my $toggle = -1;
 for (@lines) {
      my $greyline;
      if( $toggle == 1 ){ my $greyline = '' }
         else { $greyline = 'greyline' }

     $text->insert('end',"$_",$greyline);
     $toggle *= -1;
 }

MainLoop;
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