"The Hill We Climb" is overflowing with alliteration, which helps to develop its themes of unity and struggle (and also simply makes the poem sound all the more striking and memorable!). Through its music and meaning, the alliteration in this poem suggests that the speaker dreams of a more harmonious country—one that's worth fighting for.
For instance, take a look at the woven alliterative sounds in this passage from the middle of the poem:
We seek harm to none and harmony for all.
[...]
That even as we grieved, we grew.
That even as we hurt, we hoped.
That even as we tired, we tried.
That we’ll forever be tied together, victorious.
Here, alliteration draws striking connections between contrasting ideas: harm and harmony, grief and growth, hurt and hope, being tired and trying all the same. The repeated initial sounds here underline the speaker's big point: the difficulties and the triumphs of progress go hand in hand. (Also note that these lines are clear examples of parallelism and antithesis!)
Towards the end of the poem, meanwhile, the speaker uses alliteration to make her hopeful vision of the future in lines 47-49 sound rapturously musical:
If we merge mercy with might, and might with right, then love becomes our legacy and change, our children’s birthright.
So let us leave behind a country better than the one we were left.
With every breath from my bronze-pounded chest, we will raise this wounded world into a wondrous one.
The gentle but insistent alliterative /m/, /l/, /ch/, and /w/ sounds here mirror the speaker's hope for a "legacy" of tender "love." But that legacy won't come without work, and the bold /b/ sounds in "behind" and "better" ring with the armored strength of the speaker's "bronze-pounded" chest.