The repetition of consonant sounds at the beginning of closely connected words, as in 'Peter Piper picked a peck of pickled peppers,' is called

Alliteration is the repetition of the same initial consonant sound in neighbouring or closely connected words. Assonance is the repetition of vowel sounds, consonance is the repetition of consonant sounds within or at the end of words, and onomatopoeia refers to words that imitate the sounds they describe.