A little black thing among the snow, Crying "'weep! 'weep!" in notes of woe! (1-2)
The rhyme on "snow" and "woe" is a neat one, because we associate the whiteness of snow with goodness, not the evil blackness of the chimney. The rhyme, however, reminds us that part of the child's sadness has to do with the fact that he is out in the cold, both literally and metaphorically, in that his parents don't seem to care about him—at all.
"Because I was happy upon the heath, And smiled among the winter's snow (5-6)
The past tense is important; the chimney sweeper defines his sadness indirectly, by saying he used to be happy, back then, before his parents made him wear the "clothes of death." And it turns out the whole reason they made him wear those clothes is because he was happy. How does that work?
They clothed me in the clothes of death, And taught me to sing the notes of woe (7-8)
Parents are supposed to make their children happy, or at least try to; in the poem, they seem to cause nothing but sadness. They teach their child the "notes of woe," which is the sweeper's way of saying they have forced him into a life that is miserable. They sound like great parents, right? Oh wait.