He's been called the "Eosphoros of the Reformation", but how did he come to believe that the Bible opposes the idea that papal authority is above government authority?
There are whole books to be written on the subject of John Wycliffe's early Reformations, and even the Wikipedia article on the conflict is pretty substantial. As such, this is just a brief summary of three influences (in chronological order) that should be understood:
Duns Scotus first helped to diminish the idea of "universals". Universals - the hallmark of scholasticism - were concepts that had real essence in and of themselves. As Scholasticism declined, philosophers less willing to study the "ideal notion" of a thing and look more at the specifics of an individual. Its an overstatement to say that a scholastic didn't care about "the Pope" and only thought of "the Papacy," but Duns Scotus helped the idea that one could separate the individual from the instution.
William of Occam and his writings against papal authority. William of Occam was a philosopher who championed the idea that "Universals should not be propogated needlessly" or more simply "The simplest explanation is usually the best," probably the most succint way of describing Scotus... Occam also began to write about the abuses of power in general, and the need for a separation of church and state in particular. Later the concilliar movement would take these ideas as the bedrock of liberal democracy in general.
Wycliffe's own mendicacy Wycliffe himself, like St. Francis of Assisi before him, more or less rejected wealth and power, and therefore became increasingly vocal in his dislike of the papacy. It should also be noted that by the time of Wycliffe's death, there were two people claiming to be Pope, one in Rome, the other in Avignon, and would soon be a third in Pisa. This enabled one to remain in the church and write a little more openly. Even the modern Roman Catholic Church admits this was a real low point in the Papacy, and abuses were rife. Simony and Indulgences, and all the things Protestants still think of as "Catholic" were fairly normative at this time, and it wasn't until the 1540s that these abuses would be rectified.
This doesn't mean Wycliffe's own writings escaped the notice of the 15th Century Popes, however. In 1415, Wycliffe's remains were exhumed and burned, and he was excommunicated. Still, the relative disorder of the period gave Wycliffe space to write.