The conflict is that the man is traveling in well below zero temperatures, and he is all alone, except for his dog. The dog knows it is too cold to be out in this cold weather. The old timer at Sulpher Creek warned him not to travel alone.
Anytime your spittle crackles and freezes before it hits the ground, it is too cold to be out. Nonetheless, the man is trying to make it to camp where the boys are by six pm.
Possibly, the man might have made it if he had not stepped in water up to his knees. The ice looked solid but a flow of water bubbled just below the surface, and the man is now soaked up to the knees.
In one way, this is the deciding factor as to whether or not the man lives or dies. The man's fingers are frozen and he cannot get a match to light. There will be no fire. The man's inner conflict is fading. Nature is winning. When the man can fight no more, nature takes his life and the man dies.
The basic conflict is suggested by the title of the story. The man simply wants to build a fire. This should be easy because he has plenty of matches and there are trees all around him. He might have a little trouble finding dry wood, but that is not part of the problem. He can break off dead lower branches and can also find plenty of dry pine needles. The problem is caused by the extreme cold, and it is also exacerbated by the fact that he is a chechaquo, "a newcomer in the land." His inexperience causes him to make a couple of fatal mistakes. He becomes soaked because he steps into a pool of water hidden by ice and snow, and then he builds his first fire under a tree that dumps snow on the struggling flames and puts them out. So the conflict is not with the vast Yukon. It is not a question of fighting nature to get from one place to another, but only a question of building a little bonfire to keep from freezing to death. He can't even do that.