The first elements that establish the atmosphere of the story are actually found elsewhere: we're told it is sunset, which means the forest will be growing darker, and then his wife (Faith) says that she's especially worried about him tonight of all nights. Therefore, before we get the thoughts or description, we know tonight is special and growing darker.

When we get to the descriptions of the forest itself, Hawthorne piles on the adjectives and choose verbs that darken the atmosphere:
"He had taken a dreary road, darkened by all the gloomiest trees of the forest, which barely stood aside to let the narrow path creep through, and closed immediately behind. It was all as lonely as could be; and there is this peculiarity in such a solitude, that the traveller knows not who may be concealed by the innumerable trunks and the thick boughs overhead; so that with lonely footsteps he may yet be passing through an unseen multitude."

Look at how many words accent darkness and isolation—the trail closes behind him, all is hidden, dark, unseen, etc. So, the description essentially labels the literal and metaphorical path as evil, and sets the mood to match.

Brown's thoughts underscore this:
"There may be a devilish Indian behind every tree," said Goodman Brown to himself; and he glanced fearfully behind him as he added, "What if the devil himself should be at my very elbow!"

He turns the natural into the supernatural, and makes the world a fearful place.
Greg

How does the narrator's description of the forest and of Brown's thoughts establish the atmosphere of "Young Goodman Brown"?

At the beginning of the story, Brown notes Faith's uncertainties and encourages her to have "faith" in him. She pleads with him not to go on the journey but he replies that he must. After Brown takes his last look back at Faith, he criticizes himself for leaving her. He acknowledges that she, perhaps via a dream, intuits something dangerous about his journey on this night. He thinks: 

Methought as she spoke there was trouble in her face, as if a dream had warned her what work is to be done to-night. But no, no; 't would kill her to think it. 

Brown certainly shares his wife's fears about the journey and ends this section of thinking with " . . . after this one night I'll cling to her skirts and follow her to heaven." Brown, having seemingly convinced himself that he will successfully complete the journey and spend the rest of his life in "faith" with Faith, determines to get on with his journey, also called his "evil purpose." 

The descriptions of the road and the forest itself are full of synonyms of evil, darkness, and danger. 

He had taken a dreary road, darkened by all the gloomiest of trees of the forest, which barely stood aside to let the narrow path creep through, and closed immediately behind. 

Brown even supposes that the devil or some "devilish" Indians might be lurking. The atmosphere created by the descriptions of the forest and Brown's thoughts definitely conveys a sense of foreboding, danger, and evil. His journey is something he does not look forward to; it is something he feels compelled to do. He feels it is a necessary test of his faith. Therefore, he must embark on a dark journey in order to truly test that faith.