The alliteration affects the tone of the poem in the following way:
It creates a humorous tone because the speaker is amused by the cat even though it keeps the speaker awake.
Alliteration, also called head rhyme, is the repetition of consonant sounds at the beginning of words in order to create different effects, such as drama or even humor. Examples found in the poem:
A bump of black and white...
I find him near my head.
The pillow quakes and quivers...
The author, by using alliteration, is kindly ironic about his cat, since he both likes the animal and is disturbed by it.