You need to be a canon lawyer, which, nowadays, means you have a J.C.L. from a university with an accreditation in canon law. There are three in North America, Catholic University in Washington, DC, St. Paul’s University in Ottawa, and the Pontifical University of Mexico in Mexico City. There are also a bunch in Europe, some in South America, two in India, and three in Africa. If you want more information about what a canon lawyer is, this website is useful:
http://www.canonlaw.info/a_canonlawyersarearent.htm
I don’t know what the specifics were to be a canon lawyer in 16th century Europe, although I know that almost all of the European universities taught canon law.
In the case of Henry VIII’s annulment, the first hearings happened in front of Cardinal Wolsey, who was the Papal Legate in England, and Cardinal Campeggio, the Cardinal Protector of England (who was under secret instructions from the Pope to find a way to put the whole thing off until the Pope found a way to get out from under Charles V’s thumb.) Campeggio first tried to convince Henry to drop the annulment, and then to convince Katherine to agree to the annulment, and when that didn’t work, hearings began. I don’t know who represented Henry or Katherine at the hearings, though.
After nothing came of that, Henry sent the Earl of Wiltshire, the Archbishop of York, and the Bishop of London to the Pope to argue his case directly. Of the three of them, Thomas Boleyn, the Earl of Wiltshire (and Ann Boleyn’s father, so that’s awkward), had a bunch of diplomatic experience, but no legal training, Richard Lee, the Archbishop of York, had a doctorate of theology either from Oxford or the University of Louvain (he had studied at both), and John Stokesley, Bishop of London, had a degree from Oxford, and had been a lecturer and a fellow at Magdalen College there.