Alliteration is used throughout "A Dream Within a Dream." Note, for instance, the repeated /d/ sounds across lines 4 and 5:
You are not wrong, who deem
That my days have been a dream;
The /d/ sounds here link "days"—a word that stands in for life generally—with "dream." Life, therefore, is linked with illusion and unreality. But what's also notable is the way that the alliteration here seems to increase the riddle-like quality of the poem, as though the speaker is trying to figure out some vital puzzle about life. Sounds are repeated and fold in on one another, echoing the speaker's confused vision of reality as an illusion within another illusion.
Similarly, note the alliteration in line 18 of the words "while" and "weep." Not only is the phrase "While I weep" repeated (an example of epizeuxis), but the phrase itself includes repetition of the /w/ sound. Again this creates a sense of circularity, of reality folding in on itself; the speaker wants clarity but just repeats the same thing, returns to the same words and sounds without new understanding.
Later, the /h/ alliteration of "hold" and "hand" lends line 14 a breathy quality, and gives a sense of the speaker's exasperation—contrasting with the two hard /g/ sounds in the following line that emphasize the "golden[ness]" of the sand. And, of course, the poem's last two lines closely imitate the ending of the first stanza—copying the alliteration of /d/ and /s/ sounds found in lines 10 and 11 (now framed as a rhetorical question). The poem thus ends close to where it began, without any newfound clarity or certainty. Again, the speaker returns to the same sounds, as if repeating them will bring about some new understanding—but it doesn't.