Do you think the man will be able to reach camp safely?

Sadly, the protagonist of To Build a Fire does not reach camp safely. He dies at the end of the story.
One of the important themes in To Build a Fire is preparedness. The nameless narrator is not prepared for his journey across the Yukon. Though it is only ten miles, the freezing cold weather makes the trip impossible because the man is not adequately prepared. Any trek into desolate territory should include adequate gear. Jack London describes the protagonist early in his journey:

He was glad he was without a sled. Actually, he carried nothing but the lunch wrapped in the handkerchief. He was surprised, however, at the cold. It certainly was cold, he decided, as he rubbed his nose and face with his mittened hand. He had a good growth of hair on his face, but that did not protect his nose or the upper part of his face from the frosty air.

He is even proud of the fact that he has only brought lunch. He is aware of the cold but unaware that he needs actual protection from it. London explains that to the man, 50 degrees below zero is merely a temperature. It should mean "80 degrees of frost" and cause him to choose adequate protection.
Later, after he gets wet and fixes that immediate problem, the man thinks about the advice given to him by an old man before his journey: 

He remembered the advice of the old man on Sulphur Creek, and smiled. The man had been very serious when he said that no man should travel alone in that country after 50 below zero. Well, here he was; he had had the accident; he was alone; and he had saved himself. Those old men were rather womanish, he thought. All a man must do was to keep his head, and he was all right. Any man who was a man could travel alone.

The foolish confidence of the man is what eventually leads to his death—he chooses to ignore the warnings of those older and more experienced. Instead, he writes them off. By the end of the story, just before he dies, he says, "You were right, old fellow. You were right."
He dies peacefully as he goes into "what seemed to him the most comfortable and satisfying sleep he had ever known." The dog that accompanied him runs back toward the trail the man had lost, heading for the camp where people will be.