Different people may define "greatest accomplishments" in different ways. I'm not sure how you define it, but I'll give you some of the things William Penn did that I feel were most influential during his lifetime.
William first became aware of the Quaker faith as a teenager and encountered it again during his time at Oxford. He converted to Quakerism in 1667, which was against the law in the British Empire of that time. Between 1667-1670, Penn was imprisoned three times for his activities in the Quaker religion and authored several published works arguing against religious intolerance. In 1668, he became a Quaker minister.
The Tolerance Act passed in England in 1669 legalized the Quaker faith, although there was still much pressure against members of the faith. Penn helped to create the charter allowing a group of Quakers to settle in East Jersey in 1675; in 1681, he and a group of investors purchased the colony of East Jersey. In 1682, Penn took payment of a large debt owed his family by the English monarchy in the form of the grant of a large parcel of land in the American colonies. Penn signed one of the very few treaties with the Native American residents of the land he was taking to establish the colony of Pennsylvania and planned for a colony that would be free of religious persecution.
With a change in the monarchy in England, Penn had to return to England to fight for the position of Quakers. In 1692, Penn lost his rights to the colonial property. He regained the Charter in 1699, but the colonists continued to fight the use of colonial funds for military purposes (such as advocating independence for the new colony).
In 1701, Penn helped to develop the Charter of Privileges, which gave elected representatives in assembly legal powers in the colony. Penn left the colony later in 1701, battling those who wanted the colony under British governmental control and facing increasing debt. Penn was sent to debtor's prison in 1707. He died in 1718.