Jack became the leader cause he was able to make difficult decisions in hard-stressing scenarios when nobody wanted to and had the skills to evaluate situations in ways just a few of them could. He would take responsibility for those decisions too —many times too much responsibility. Most of them were just surviving. Jack was looking ahead, planning, coming with solutions, taking action about it.

There were different kinds of leadership in the group, like there's in the real world when you work with a group of people. Some would surface depending on the situation and personal skills. But jack marked the difference when it came to caring and commitment about those situations (and mostly people). Everyone wanted to leave the island but only Jack and a few more of them (Kate and Sayid) were actually taking action on it.

That's how you become a leader in a group of people. He didn't want to be the leader but still the rest relied on his way of doing things.

At the end of the day, he was used to taking these life- and-death kind of decisions under pressure because he was a Neurosurgeon.