Gotta validate the inputted word by checking that it indeed consists of the chars in $random_string.
Perhaps I'm misunderstanding you, but this sounds like you want to validate that the two sets of string characters are equal (the 'compare uniquely')--not that the two strings are equal. If this is the case, the following is an option:
my $word = 'FHDJSKAL';
my $random_string = 'ALSKDJFH';
if ([sort ("\U$word" =~ /./g)] ~~ [sort ("\U$random_string" =~ /./g)])
print "Good word.\n";
print "Try again.\n";
Hope this helps!
Update: Replaced the hash solution with one using the smart operator (~~), as the former didn't match properly.