Alliteration is used throughout "They Flee From Me." In the first two lines, for example, soft /f/ sounds combine with sibilance to create a hushed atmosphere:
They flee from me that sometime did me seek
With naked foot, stalking in my chamber.
The /s/ sound has a particularly whispery quality to it, suggesting that there was something dangerous or illicit about the romantic meetings the speaker here recalls. That quietness fits with the idea of a predator "stalking" its prey, patiently waiting for the right moment to strike.
In the second stanza, the speaker remembers a cherished night that he spent with his ex-lover. Life was "Twenty times better" than it is now, the speaker says, the alliteration working to intensify this sentiment (which can be read as a kind of exaggeration or hyperbole); that double /t/ sounds makes the phrase stand out all the more strongly for the reader.
Later in the same stanza, the poem returns to the sibilant /s/ sound to suggest passion and intimacy:
And she me caught in her arms long and small;
Therewithall sweetly did me kiss
And softly said, “Dear heart, how like you this?”
The breathy /h/ sounds and additional consonance of /s/ sounds contributes to the effect here, evoking a hushed—but erotically charged—atmosphere.
Another example of alliteration appears in line 17:
Into a strange fashion of forsaking;
Here, the speaker is wondering if his lover's change of heart is part of a new "fashion of forsaking." That is, he suspects that there is a new trend in town, one which involves no longer remaining loyal to him in order to pursue other love interests. He sees it as a kind of artifice or pretension, which he struggles to reconcile in his mind with the emotional and physical intimacy of the moment described in the second stanza. Alliteration is one way in which the poem can perform this kind of artifice, reminding the reader that the poem is something constructed and deliberate. The /f/ is sound is strikingly visible, like a new style of hat that everyone suddenly seems to be wearing around town.
The poem's penultimate line turns the earlier /s/ alliteration on its head:
But since that I so kindly am served
These /s/ sounds have an embittered quality, as though the speaker is talking through gritted teeth.