Learn the difference between alliteration and assonance; these two are commonly used tools which a poet has at his disposal and they are easily confused. Whilst alliteration relates to the same consonant letter or sound at the beginning of a word, assonance relates to the pairing of vowel sounds. So whilst "Peter Piper picked a peck of pickled pepper" is a well-known example of alliteration, so "the rain in Spain falls mainly on the plain" is a well-known example of assonance.
It is vital to understand this key difference between these two writing techniques before attempting to find examples of alliteration in poetry, because without a clear understanding then it is quite easy to get the two methods confused.
Read through the poem from start to finish, following the flow of the poem with pen or pencil. Where you encounter an example of alliteration, or notice many words close together that start with the same letter, underline the initial letter of the words which you feel are examples of alliterative poetry.
Continue with this process until you have finished the poem, and then go back to the beginning and review the letters which you have underlined to ensure that they match the definition of alliteration. Be very clear in your mind of the difference between alliteration and assonance as per the first step.
Read the poem aloud. Sometimes the letters or sounds in poetry don't have the same impact when they simply read silently on the written page. To speak the words aloud will add emphasis and resonance to the beginning of each word and can be a very effective aid for finding alliterative lines in poetry.
It is still worth having your pen or pencil handy whilst reading the poem aloud so that you may continue to underline any new alliterative words that you discover.