Alliteration is used quite frequently in "To an Athlete Dying Young," usually to reinforce the poem's thematic points or to create a sonic echo of its literal meaning.
In the first stanza, for example, it links "chaired" with the word "cheering." Here, the speaker is recounting a memory in which the townspeople carried the athlete—victorious in a race—through the streets. The alliteration links the words through sound to reinforce that they are linked in meaning too. Both are meant to conjure a sense of celebration, and tying them together by alliteration shows that they are part of the same occasion.
Line 5 uses alliteration in the phrase "road all runners come." Here, the shared /r/ sound creates a sense of exertion and physical effort. The two /r/s are separated by just one syllable, making the letter sound persistent, as though it is determined to exert its presence on the line.
Line 19 reintroduces this alliterative /r/ sound, again linked with the act of running. Three /r/s make the sound even more persistent, suggesting desperation (though the third is consonance, not alliteration). This is likely intended to evoke the inevitability of what it describes: all "runners" who live to an old age will be "outrun" by "renown:" fame will escape them. Both instances bring to life both the runners' physical efforts and, perhaps, the broader idea of humans doggedly trying to outrun death.
In line 13, "shady" and "shut" alliterate. Here, the speaker is talking about the athlete's eyes, now closed in death. Both words are associated with darkness, and the fact there is a pair of matching sounds perhaps represents the two closed eyes.
In lines 21 and 22, the poem discusses the athlete's echoing footsteps. To bring this to sonic life, /f/ sounds are placed throughout both lines, creating a linguistic echo to match the lines' literal meaning.
Finally, the last line links "garland" with "girl's." Here, the alliteration feels like a decoration, a way of reinforcing the image of a garlanded young girl. The image of a pretty child, which is so different from that of the striving athlete, makes the athlete's garland seem suddenly meaningless, and the alliteration reinforces the idea that his accomplishments might not matter much in the underworld.