Unless I'm missing something, the program that creates this file is already counting the hops for you. Each section starts with a line that begins with Traceroute:, followed by the number of hops plus 1, and ending in the destination IP address. That being the case, I'd just go through the file watching for those lines; no counting necessary:

#!/usr/bin/env perl
use Modern::Perl;

my $count = 0;
while(<DATA>){
    if( /^Traceroute: (\d+) .+ (\d+\.\d+\.\d+\.\d+)$/ ){
        my $h = $1 - 1;
        say "$h hop". ($h==1 ? '' : 's') . " to $2";
        $count++;
    }
}
say "---  $count traces found  ---";

__DATA__
Traceroute: 11 1291796822 1291796821 1291796821 OpenDNS 0.chstatic.cvc
+dn.com 216.137.61.109
traceroute to 216.137.61.109 (216.137.61.109), 30 hops max, 60 byte pa
+ckets
 1  192.168.178.1  0.638 ms  7.361 ms  7.366 ms
 2  82.135.16.28  44.925 ms  46.633 ms  50.457 ms
 3  212.18.7.121  98.010 ms  98.307 ms  98.650 ms
 4  212.18.6.97  68.374 ms  72.222 ms  76.184 ms
 5  62.140.24.49  129.246 ms  130.224 ms  130.221 ms
 6  4.69.134.2  98.205 ms  105.207 ms  102.287 ms
 7  4.69.140.10  106.816 ms  82.609 ms 4.69.140.2  88.631 ms
 8  4.68.23.76  79.559 ms  65.093 ms 4.68.23.204  64.530 ms
 9  212.162.24.70  65.476 ms  50.797 ms  55.108 ms
10  216.137.61.109  58.221 ms  62.351 ms  59.094 ms
Query: 18 1291796822 GoogleDNS 0.chstatic.cvcdn.com 1 True 0.155671834
+946 0.0507638454437
Traceroute: 2 1291796823 1291796822 1291796822 LocalDNS LocalDNS 192.1
+68.178.1
traceroute to 192.168.178.1 (192.168.178.1), 30 hops max, 60 byte pack
+ets
 1  192.168.178.1  0.649 ms  1.033 ms  1.013 ms
Query: 13 1291796823 LocalDNS 0.media.collegehumor.com 1 True 0.193642
+139435 0.0472960472107
Traceroute: 9 1291796827 1291796822 1291796822 GoogleDNS GoogleDNS 8.8
+.8.8
traceroute to 8.8.8.8 (8.8.8.8), 30 hops max, 60 byte packets
 1  192.168.178.1  0.464 ms * *
 2  82.135.16.28  68.613 ms  72.079 ms  76.268 ms
 3  212.18.7.101  160.966 ms  163.495 ms  163.517 ms
 4  93.104.240.53  99.638 ms  103.396 ms  108.241 ms
 5  209.85.241.110  111.919 ms 209.85.240.64  114.900 ms 209.85.241.11
+0  119.654 ms
 6  209.85.254.116  124.134 ms 209.85.254.118  127.998 ms 209.85.254.1
+12  111.621 ms
 7  209.85.249.166  116.258 ms 209.85.254.126  115.774 ms 209.85.254.1
+34  105.670 ms
 8  8.8.8.8  95.428 ms  95.573 ms  95.923 ms
Query: 10 1291796827 OpenDNS 1.im.cz 1 True 0.0573320388794 0.05522108
+078
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