"She Walks in Beauty" makes frequent use of alliteration. For the most part, the alliteration is meant to reflect beauty on a linguistic level. In other words, the alliteration helps the poem itself sound beautiful to match the fact that it is talking about beauty. Indeed, it's quite ornate, even showy at times. It draws the reader's attention to the fact that this is capital-P Poetry.
The first example of alliteration is in line 2: "cloudless climes and starry skies." The alliteration here is not subtle, and forms two obvious pairs—the hard /c/ sound and the /s/ sound (which is specifically an example of sibilance). As the poem intends to establish beauty as a kind of perfection achieved through harmony (in this case, between light and dark), the pairing-off of sounds creates a sense of harmoniousness; in other words, the sounds—two /c/ sounds in a row followed by two /s/ sounds in a row—create a sense of balance and symmetry.
The next key example is between lines 4 and 5, with "Meet" and "mellowed." "Meet" is a stressed syllable, which opens the line with a trochee and upsets the poem's iambic tetrameter:
Meet in her aspect and her eyes
This stress makes the word "Meet" sound particularly prominent, allowing for the alliteration between this and "mellowed" to ring out despite the relatively wide distance between these two words. "Mellowed" in turn shows the reader what happens when the "best of dark and bright / Meet" in the appearance of the woman. As such, cause is linked with effect through sound.
Line 9 has alliteration through the phrase "Which waves," which is part of the stanza's broader heavy use of consonance, assonance, and sibilance. All of these techniques function as ornamentation, intended to make the stanza sound "prettier" in order to reflect the woman's beauty.