Pope John II was the successor of Boniface II, who had originally chosen the deacon Vigilius to be his successor.
During the reign of Boniface the people of Rome gave Pope Boniface the right to choose his own successor. Vigilius, who would later use the money and influence of the empress Theodora to obtain the papacy, was apparently a poor enough choice that there was an outcry against the practice by the very people that had given Boniface the right!
As a result, Pope Boniface II died without a successor, and after a period of two months Pope John II ascended. The situation at that time is described in this way by the Catholic Encyclopedia (note that is the Roman Catholics, not Protestants, saying this):
Simony is the purchase of clergical positions, and Roman Catholic leaders were fighting the practice for at least the next 1,000 years. Pope Urban VI, for example, called his own court traitors to Christ because they had purchased positions, then abandoned those positions to hang out in Rome.
It's not known how Pope John II obtained the papal chair. It is known that the favor of kings was important at that time. Rome had no emperor in the west. Emperor Justinian I was ruling from Constantinople—but according to historians he himself was under the rule of his wife Theodora—and the Goths occupied Italy. Both monarchs had favor for John, and that would have ensured his appointment to the position.
His successors were not so fortunate. The Gothic king had Silverius installed after Agapetus I ruled for less than a year. Mere months into Pope Silverius' reign, the Empress Theodora send the commander of the Byzantine army, Belisarius, into Rome to install Vigilius as pope, hoping to see the patriarch Anthimus reinstalled in Constantinople. Despite the fact that King Athalaric besieged Rome as this was happening, Empress Theodora succeeded, and Vigilius ensured that Silverius was ill-treated enough to die in prison after a few months.
The circumstances of Pope John II's death are not known, but he reigned for only two years, dying on May 8, 535.
The Catholic Encyclopedia says it's likely Pope John II was the first pope to change his name upon becoming pope. I will quote them directly on this, as Philip Schaff says nary a word about it, and I had a hard time finding anything but a sentence or two given to this pope otherwise.