In Hatchet by Gary Paulsen, what is the most important thing Brian does to survive?

The most important thing that Brian does to survive in Hatchet is develop his mental toughness.

Brian was in a state of panic prior to recognizing the power of mental strength.  He had taken over flying and landing the plane after the pilot suffered a heart attack. The crash landing was in the middle of the wilderness, severely wounding Brian. Hailing from an urban area, Brian had no real idea of what to do.  He found himself struggling for food and shelter, while being victim to animals and natural elements.  All of these realities begin to take a toll on him.  Anxiety settles in, and his survival chances decrease as a result.

Brian has to develop mental strength to confront the challenges that confront him.  Developing this element is the most important thing he does to ensure his survival. One of the first moments where we see this is in Chapter 5.  In the midst of his tension, Brian remembers the instruction of one of his teachers. This helps him establish the mentality needed to survive challenging situations:

Brian had once had an English teacher, a guy named Perpich, who was always talking about being positive, thinking positive, staying on top of things. That's how Perpich had put it—stay positive and stay on top of things. Brian thought of him now—wondered how to stay positive and stay on top of this. All Perpich would say is that I have to get motivated. He was always telling kids to get motivated.

Thinking about Perpich's words triggers the first instance of Brian figuring out how to survive.   Instead of dwelling on all of the things that have gone wrong or everything he does not have, Brian "gets motivated" about what he needs to do.  After thinking about Perpich's teachings, Brian identifies the need for shelter and food, and uses his shirt to keep the "vampires" or mosquitoes off his back. The ending to the chapter shows the value of Perpich's instructions:  "I have to get motivated, he thought, remembering Perpich. Right now I'm all I've got. I have to do something."

When Brian establishes the mental toughness to face down the elements around him, his survival chances increase.  Things begin to fall into place for him.  He gets his first bit of luck when he throws the hatchet against the cave wall, making sparks that allow him to figure out how to create a fire.  From this, Brian figures out how to find food and cook them over the fire.  He enhances his shelter, creating a food shelf and a door to keep out intruders.  Brian develops a toughness that enables him to survive attacks from a moose as well as a tornado.  Brian maximizes his chances for survival by developing mental toughness.  This enables him to endure the natural elements, and invariably triumph over them.