In terms of who, I mean what was his position and relation to previous Caliphs. As I understand it, he was neither one of Mohammed's Companions or close relations. So what did he have going for him that he managed to ascend to the Caliphate after Ali's assassination?
Muawiya was actually one of the Companions (Sahaba), although a younger and definitely not senior one. He was the son of Abu Sufyan, the chief of the Banu Abd Shams and essentially the leader of the Quraysh before his conversion to Islam after the Conquest of Mecca. His sub clan was Banu Umayyah(of which Uthman, his cousin and the third Caliph, was also a member) hence the Umayyad Dynasty.
He was a katib-e-wahi (writer of Revelation) and the Prophet's scribe. His sister Umm Habibah was married to the Prophet and so it is reported that the Prophet named him the "Maternal uncle of the Believers".
After the Prophets death, he participated in several battles and eventually became governor of Syria. He was also behind the formation of a navy and strengthening the military.