How do I approach an unseen poem?

It is very important to remember when analysising an unseen poem that is NOT PROSE. This means the tools that you apply to the text must take into account the differences between the two. To make sure you distinguish your approach to poetry from your approach to prose, it is most important that you focus on FORM, VOICE and SOUND.
An essential component of poetry is FORM. How many stanzas are there? How long are they? What is their relation to each other? These are all questions that focus in on the form of the poem, as they interpret the thinking behind how the poem is structured. Are lengthy stanzas placed beside shorter ones merely by chance? Or is that a deliberate choice by the poet to weigh short, sharp clarity against longer, more aimless strands of the poem?
Noticing the difference between stanzas, and how they relate, is also crucial to understanding VOICE in poetry. Poets often play with the concept of voice, challenging your assumption that there is one speaker throughout. The 'I' from stanza 1 may not be the 'me' in stanza 2. The question as to who is speaking in any given poem is very rarely a straightforward one. Who are these voices? Where are they speaking from, and what are their motivations? Think of these voices as different masks, or characters, of the poet's voice. They may not be distinct personalities, but they may show different tones, dictions or accents which you can identify.
When analysing what these voices are actually saying, it is important to remember that poems are meant to read aloud, meaning that SOUND plays a crucial role. Does the poem use rhyme, and to what effect? Are they obvious nursery rhymes that chime together easily or are they more are they disjointed half rhymes that share some sounds but not all? Do the words look like they rhyme, like 'raid' and 'said', but actually don't? Are there internal rhymes in the poem and how does this affect how you read it? There are other sound techniques used by poets that you should look out for. Assonance (shared vowel sounds, as in the 'o' sound in 'row' and 'mouth') and Consonance (shared consonant sounds, as in the 'p' in 'help' and 'stop') are also important to look out for, especially if the poem does not use rhyme (or at least not obviously).
Poetry is an art which like scultpure has shape, like drama has characters and voices, and like music has sound. These elements must be incorporated into any analysis of poetry.