Alliteration is used to powerful effect in "Anthem for Doomed Youth."
The first example is in line 3, and this is perhaps one of the most famous examples of alliteration in English poetry. Indeed, it is often used to illustrate the definition of the poetic device:
Only the stuttering rifles' rapid rattle
This line uses the repeated /r/ sounds (including the consonantal /r/ in "stuttering") to emulate the sound of relentless gun fire. The alliteration is deliberately heavy and obvious, calling to mind the hail of bullets that WWI soldiers had to face when they went out of the trenches. As the stanza talks specifically about the horrible sounds of warfare—contrasted with the pleasant but meaningless "passing-bells" in churches back home—it makes sense for the poetic "volume" to be turned up so loud with this alliteration.
The next example is in line 5:
No mockeries now for them; no prayers nor bells;
These /n/ sounds are linked with the concept of negation—that is, the closing-down or rejection of possibilities. Essentially, they link the words "no" (and "nor") with the word "now," subtly pointing out that for those who have died in the war there is no now—they are no longer part of the present, but rather part of the past.
Lines 7, 8, and 11 also have alliteration, but that specific type is covered in the sibilance section. Line 12 ties "pallor" and "pall" together through similarity of sound. This is part of the line's metaphor that, rather than a flag draped over their coffins, the dead young men will be honored in the grief-stricken faces of their loved ones (in particular "girls" like their sisters, wives, and girlfriends).
The last line also features alliteration, with the /d/ sounds of "dusk" and "drawing-down" combining with gentle consonants to bring the poem to a hushed close (in contrast to the bombastic music of national anthems, which the speaker has previously made clear is an inappropriate way to honor soldiers' deaths).