If you have a limited sequence of characters following one another you can use a Pattern, .find() on a matcher on your input and just invert the test:
// Only the alternation is needed, no need for the capture
private static final Pattern PATTERN
    = Pattern.compile("012|123|234|345|456|567|678|789|890");
// ...
if (PATTERN.matcher(input).find())
    // fail: illegal sequence found
But if you want to detect that code points follow one another you have to use character functions:
final CharBuffer buf = CharBuffer.wrap(input);
int maxSuccessive = 0;
int successive = 0;
char prev = buf.get();
char next;
while (buf.hasRemaining()) {
    next = buf.get();
    if (next - prev == 1)
        successive++;
    else {
        maxSuccessive = Math.max(maxSuccessive, successive);
        successive = 0;
    }
    prev = next;
}
// test maxSuccessive
Note however that this will test successive characters according to "canonical ordering", not collation. In some locales, for instance, what is immediately after a is A, not b.
More generally, if you want to test for password requirements and constraint evolves, you are better off splitting things a bit. For instance, consider this:
public interface PasswordChecker
{
    boolean isValid(final String passwd);
}
Implement this interface for each of your checks (for instance, length, presence/absence of certain characters, etc), and when you do check for a password, have a List of checkers; the password is invalid if one checker returns false:
private final List<PasswordChecker> checkers = ...;
// then
for (final PasswordChecker checker: checkers)
    if (!checker.isValid(passwd))
        return false;
return true;
If you use Guava, you can forget about PasswordChecker and use Predicate<String>.