After their deaths, do popes become saints? I’d think that, having attained such a lofty position, sainthood would be a given. Or do they need to have performed miracles?
First, a clarification: A saint is any person who’s in Heaven, and the Church has never definitively stated that anyone is not in Heaven. So far as any of us mortals know, Hitler, Stalin, and Nero could all be in Heaven. A canonized saint is just a person for whom the Church has officially said that they know that person is in Heaven.
That said, some Popes are canonized, some are not. The requirements for canonization of a Pope are the same as for canonizing anyone else. They do, of course, have a big advantage, in that they’re well-known, and usually widely regarded (within the Church, at least) as having been righteous men. It would be not at all surprising if John Paul II were to be canonized, but it’s not guaranteed.
The most recent Pope to be made a saint (at least in terms of recency of his papacy) is Pope St. Pius X. He was pope from 1903 until 1914 and was canonized in 1954
In the past 1000 years, only five popes have been sainted, and ten more have added “Blessed” to their title, a step on the way to sainthood. So it is unusual for popes to be saints (though it happened often in the early years of the church – there are 35 saints as pope before there was one who wasn’t sainted).
Essentially, a pope has the same qualifications for sainthood than any other Catholic. And, like any other saint, there have to be miracles attributed to them, among other things.
This is at least in part because virtually all of the early Popes were martyred in persecutions by the Romans, and martyrs are usually considered to be “automatically” saints. The first non-saint pope was Liberius (352 AD), who became pope after Constantine allowed Christianity in the Empire. However, almost all popes continued to be made saints up to the early 500s.
Liberius gets a lot of good-natured ribbing from his fellow ex-Popes up in Heaven: “Ha, Ha, You’re not canonized yet!” 
Seriously, the Church Universal has always had a tradition that identification of sanctity and hence sainthood by popular acclaim is the starting point, which is then officially recognized by an authoritative body. This is still the process in Orthodoxy, where U.S. saints include St. Herman of Alaska and St. Tikhon, who is patron of New York City in view of his many years there. Anglicanism has never officially recognized a saint by a true canonization process, but every national Anglican church celebrates minor feast days for its late national leaders. Phillips Brooks, Bishop of Massachusetts and author of “O Little Town of Bethlehem,” is a typical example, as is C.S. Lewis, and Kamehameha and Emma, the Hawaiian royal couple who converted to Episcopalianism and led a conversion in the Islands.
Catholicism has formalized the process a bit more than other churches. There is a Congregation for the Causes of Saints before which “causes” [public movewments and petitions] for canonization of a person are brought. The Canonization Process in Catholicism.
An anecdote: On Saturday, April 2, I was listening to Relevant Radio (a Catholic radio station). One caller related a short story. Two priests were walking and talking in the halls of the Vatican. They were talking about Padre Pio (who was not canonized by then). One of them accidently called him “Saint Pio.” The other priest said, “He’s not canonized yet.” Unbeknownst to them, the Pope was behind them. (I think she said it was Pope Paul VI.) He said, “He’s not canonized, but you can call him ‘Saint Padre Pio’.” Which is why some Catholics are already calling the Holy Father “Saint John Paul the Great.” As a matter of fact, a number of people on the radio confessed to have already begun asking for his intercession. One show host basically said that when his wife informed him of the news, he sat down. Once he gathered his thoughts, he prayed for the Pope and then asked for the Pope’s intercession.
Pope John Paul the Great beatified and canonized more people than any of his predecessors. I think he wanted to teach us something about saints: there have been and are more saints around than maybe what we may have thought before. In every walk of life, in every place of the world, there have been and there are saints. By elevating so many people, he elevates all of humanity by showing us just how common saints actually are. This gives us hope in humanity, and shows forth God’s redeeming and transforming grace.
WRS
As mentioned, Papacy does not automatically put you on the path to canonization. In modern times, you are expected to have displayed Christian virtue to a heroic degree in either your life or your death (martyrs, IIRC get a “bye” on some of the procedural requirements) to even be considered.
Canonization, in the RCC, means that this “saint” is included in the “canon”: there is a Feast Day and a specific Service is authorized to be performed in his/her honor. This is commonly translated as “we are pretty sure this one IS in Heaven”.
JP2 should easily get in on the ground floor with the rank of “Venerable”, if only because he will be venerated (in Krakow and Wadowicje the shrines could well be going up as we write). Beatification and canonization OTOH will have to meet the usual chain of requirements – including a 5-year cooling-off period and a series of miracles.