The major connection between Pope John Paul II and the Cold War came from the fact that the Pope was Polish by birth. Poland was a communist country but one in which there was a strong devotion to Catholicism. This was a source of tension in the country because communists are supposed to be atheistic.
When John Paul II became pope, he tried to help bring about reform in Poland. His election as pope galvanized Polish Catholics and helped encourage them to push for these reforms. An example of this was the creation of the Solidarity union, which was backed by the pope and by many of the Polish people and eventually helped to bring down communism in Poland.
So, Pope John Paul II was connected to the Cold War in that his election as pope helped to weaken communism in Poland and, thereby, in the entire communist bloc.