They were selected by a committee.
The Philosophical Institute of Victoria was founded in 1854. In 1857 the Philosophical Institute formed an Exploration Committee with the aim of investigating the practicability of fitting out an exploring expedition.
The Exploration Committee called for offers of interest for a leader for the Victorian Exploring Expedition. Only two members of the Committee, Ferdinand von Mueller and Wilhelm Blandowski, had any experience in exploration, but due to factionalism both were consistently outvoted.
Several other people were considered for the post of leader and the Society held a range of meetings in early 1860. Burke was selected by committee ballot as the leader, and Wills was recommended as surveyor, navigator and third-in-command. Burke had no experience in exploration and it is strange that he was chosen to lead the expedition.
Burke was an Irish-born ex-officer with the Austrian army, and later became police superintendent with virtually no skills in bushcraft. Wills was more adept than Burke at living in the wilderness, but it was Burke's leadership that was especially detrimental to the mission.