How to define custom exception class in Java, the easiest way?

No, you don't "inherit" non-default constructors, you need to define the one taking a String in your class. Typically you use super(message) in your constructor to invoke your parent constructor. For example, like this:

public class MyException extends Exception {
    public MyException(String message) {
        super(message);
    }
}

A typical custom exception I'd define is something like this:

public class CustomException extends Exception {

    public CustomException(String message) {
        super(message);
    }

    public CustomException(String message, Throwable throwable) {
        super(message, throwable);
    }

}

I even create a template using Eclipse so I don't have to write all the stuff over and over again.