How does Jack become the chief?

Basically, the group breaks up over prioritization. Ralph, Piggy and Simon believe that The Fire and being rescued is the most important thing. While, Jack believes that its no big deal if the fire goes out as long as they can hunt. That would be the immediate reason.
But when you read the novel thoroughly, the strains in the Ralph and Jack relations are evident right from the part where Ralph becomes chief.

Due to difference in opinions, Jack breaks away from the group and forms his own at the other side of the island. To this group, he is The Chief.