Late to the party, again, but one thing nobody has commented on is that your if-structure re-tests the same thing multiple times. Instead of:

if (($genome[$i] eq 'a') && ($genome[$i+1] eq 'a'))    { ++$tt; }
         elsif (($genome[$i] eq 'a') && ($genome[$i+1] eq 'g')) { ++$a
+g; }
         elsif (($genome[$i] eq 'a') && ($genome[$i+1] eq 'c')) { ++$a
+c; }
         elsif (($genome[$i] eq 'a') && ($genome[$i+1] eq 't')) { ++$a
+t; }
         elsif (($genome[$i] eq 't') && ($genome[$i+1] eq 'a')) { ++$t
+a; }
         elsif (($genome[$i] eq 't') && ($genome[$i+1] eq 'g')) { ++$t
+g; }
         elsif (($genome[$i] eq 't') && ($genome[$i+1] eq 'c')) { ++$g
+a; }
         elsif (($genome[$i] eq 't') && ($genome[$i+1] eq 't')) { ++$t
+t; }
[download]

you should have had:

if ($genome[$i] eq 'a') {
    if    ($genome[$i+1] eq 'a') { ++$tt; }
    elsif ($genome[$i+1] eq 'g') { ++$ag; }
    elsif ($genome[$i+1] eq 'c') { ++$ac; }
    elsif ($genome[$i+1] eq 't') { ++$at; }
}
elsif ($genome[$i] eq 't') {
    if    ($genome[$i+1] eq 'a') { ++$ta; }
    elsif ($genome[$i+1] eq 'g') { ++$tg; }
    elsif ($genome[$i+1] eq 'c') { ++$ga; }
    elsif ($genome[$i+1] eq 't') { ++$tt; }
[download]

etc.

Of course, the hash is a better solution for this problem, but I thought that the redundant testing problem should be pointed out.