The poem uses alliteration to bring its images to life on the page. There isn't a ton of alliteration here, however, which makes the moments when it does pop up all the more memorable.
In line 2, for example, the shared /m/ of "Minister manqué" adds a little poetic flourish as Hughes describes the image of Otto as a lapsed Lutheran minister. The alliteration at the start of the second stanza is even more striking. Note the loud, percussive /b/, /p/, and /c/ sounds:
A big shock for so much of your Prussian backbone
As can be conjured into poetry
The sharp sounds here add to the portrayal of Otto as a stern and proud individual (Plath's own poetry criticizes her father for this aspect of his personality). These sounds, in other words, bring that "Prussian backbone" to vivid life. (Also note that the consonance of this passage adds to the effect, via the hissing and sharpness of "shock," "Prussian," and "backbone.")
Of course, Hughes implies throughout that this isn't really Otto, and is instead a version of him that has been conjured into being on the page, a kind of phantom made out of poetry. The alliteration here thus subtly reminds readers that they're reading a poetic description of Otto, encountering the mythical idea of the man that Plath created in her work.
The other main example of alliteration comes in line 19, in which Hughes says to Otto:
This underworld, my friend, is her heart's home.
This "her" refers to Sylvia Plath. The suggestion here is that the underworld—the kingdom of death—is where Plath longed to be in life. That alliteration suggests a melding between Plath herself and death.