Do I have to pay someone to use Shakespeare characters in my own production?

There is no protection under US copyright law that would still cover Shakespeare's works. They fall in the public domain. However, you still cannot copy someone else's work that uses Shakespeare's characters, it needs to be an original work of yours. Here is a URL that gives you guidance on copyright terms under US law: http://copyright.cornell.edu/resources/publicdomain.cfm. For copyright terms in other countries, you must find out the law in each of those countries for specifics.