You probably need to read the documentation -- although, having just looked at the perldoc for Digest::MD5, it's a little opaque.

You don't need to read anything from the file, or set the binmode .. something like

#! c:\\Perl\\bin\\Perl.exe

use Digest::MD5 qw(md5_hex);
open(FILE, "logs.txt");  # YOU NEED ERROR CHECKING HERE
$md5_hex = Digest::MD5->new->addfile(FILE)->hexdigest;
close(FILE);
print "MD5: " . $md5_hex . "\n";
[download]

will probably work fine. I just deleted unnecessary lines from your code.