Here's a quick rewrite:

#!/usr/local/bin/perl -w
use strict;

my $fichiernombre = "/cgi-bin/compteleshits";
open(HITS, "+< $fichiernombre") or
  die "Could not open $fichiernombre: $!\n";
flock(HITS, 2) or 
  die "Could not lock $fichiernombre: $!\n";
my $num=<HITS>;
seek(HITS, 0, 0);
$num++;
print HITS "$num";
close(HITS) or 
  die "Could not close $fichiernombre: $!\n";
[download]

This is a rough, yet workable, version of what you want. Note that if you ar doing anything but simply incrementing the number in the file, you should add a 'truncate' command in there as well. See my tutorial on File Locking to understand *why* you need file locking.