By assigning a file handle to a scalar you are only reading one line from it. To solve the problem you just need to put your code in a loop.

Code snippet below should give you some idea

while (my $genome= <FH>) {
    for (0..length($genome)-2) {
        my $dimer = substr($genome, $_, 2);
        $dimers[$idx]{$dimer}++ ;     #seq wise length of dimer
    }
    $idx++;
}
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Update: Fixed typo. Anyhow post seems similar to of flexvault :)