Assuming that you expect the answer to be '25:15:39', and just for the sake of it - here is a solution without using any module:

#!/usr/bin/perl -wl
use strict;

my $time1 = '12:30:36';
my $time2 = '12:45:03';

print sum_times($time1, $time2);

sub sum_times {
    my $secs;
    for (@_) {
        chomp;
        my ($h, $m, $s) = split /:/;
        $secs += ($h*3600 + $m*60 + $s);
    }
    return sec_to_hms($secs);
}

sub sec_to_hms {
    use integer;
    local $_ = shift;
    my ($h, $m, $s);
    $s = $_ % 60; $_ /= 60;
    $m = $_ % 60; $_ /= 60;
    $h = $_;
    return sprintf("%02d:%02d:%02d",$h, $m, $s);
}
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Notes:

• Does not perform any sanity checking on the data.
• Should "add" as many "times" as you wish to throw at it.
• You are still probably better advised to use a module for this type of task.

Cheers,
Darren