Alliteration is used throughout "The Chimney Sweeper." The first example is in line 2:
Crying "weep! weep!" in notes of woe!
The repetition of "weep" (technically known as epizeuxis) chimes with the /w/ sound of "woe" to link the act of crying with the chimney sweeper's state of misery. This helps establish the poem's air of tragedy, a tragedy that seems easily preventable—yet, for the sweepers themselves, painfully inevitable. Without the help of adult authorities, there is little they can do to change their fate.
The next big example is in line 5, which is part of the chimney sweeper's explanation for the causes of his pain and impoverishment:
"Because I was happy upon the heath,
These two /h/ sounds ring together playfully, fitting the image of a happy child frolicking on the heath (a kind of field). It links happiness with the outdoors, contrasting the open and free space of the heath with the claustrophobic, hellish interior of the chimneys. The following line echoes this alliteration in "smil'd" and "snow," again drawing a link between joy and the natural world.
There is gentle /th/ alliteration at the start of line 10:
They think they have done me no injury,
The gentleness of the sound subtly evokes the line's sentiment—that, from the adults's perspective, nothing wrong has been done (because they are dutiful Christians). The sound also echoes the consonance in line 3—"thy father and mother"—which helps to group together all the different failures in authority that create the horrible conditions of the chimney sweeper's life. (Note that the /th/ of "think" sounds a bit different from that of "they," because the former is unvoiced while the latter is voiced. Some might not mark this technically as alliterative, but the sounds are similar enough to create a gentle effect altogether here.)
Finally, line 11 uses alliteration in "gone" and "God." This subtly emphasizes the poem's main message, which again is about the failure and corruption of authority. This /g/ sound links "God"—specifically organized religion—with the state of being "gone" (absent and neglectful). It therefore helps the poem point a finger at the Church as the cause behind the impoverishment and exploitation of the young.