What should you consider and examine when writing an analysis of an unseen poem?

Imagine you were presented with this sonnet by Seamus Heaney as an unseen poem in your exam:
When all the others were away at MassI was all hers as we peeled potatoes.They broke the silence, let fall one by oneLike solder weeping off the soldering iron:Cold comforts set between us, things to shareGleaming in a bucket of clean water.And again let fall. Little pleasant splashesFrom each other's work would bring us to our senses.
So while the parish priest at her bedsideWent hammer and tongs at the prayers for the dyingAnd some were responding and some cryingI remembered her head bent towards my head,Her breath in mine, our fluent dipping knives—Never closer the whole rest of our lives.
When writing an unseen analysis or commentary, the first thing to do is read the poem two or three times before even thinking about what you can notice to write on. Whilst you can't read aloud in an exam, it is good to try to hear it inside your head so you can understand the way in which the poet is using sound.
When you do begin to write your response, the first thing you need to do is write a one sentence summary of the action of the poem; prove to the examiner that you understand it and know what is going on. For this poem, such a sentence could be: 'In this sonnet, Heaney depicts a moment of extreme quotidian intimacy between a mother and son which returns as a comforting image at the moment of the mother's death'.
Then, try to construct a coherent argument about the poem: what do you think it, or the poet, is doing or achieving? To construct this argument, however, you will need a lot of close reading and textual detail. You can make sure you do not neglect any aspect of the poem by remembering to consider its form and structure, genre, language, imagery, and sound. Once you have looked at the poem for all of these, you will have lots of material for your argument.
Initially, then, we can consider the form of Heaney's poem. It's evidently a sonnet - a form which has inescapable romantic connotations - and yet it is about the speaker's mother; what does this relationship between form and subject matter reveal? Is the speaker romanticising his own childhood? The mundane existence of every day life? It has a 'volta' - a change in focus - in-between the octave and the sestet. What does this shift reveal? Is Heaney paying homage to generic conventions or is he rather subtly subverting them?
Sonnets traditionally have very defined rhyme schemes, but Heaney's poem revels in half-rhymes and internal rhymes; what does this deviation from tradition reveal? Does the poem feel more delicately balanced because of it? The last two lines are a couplet: Heaney formally and aurally reinforces the image of the pair existing and working together. Rhyme can be linked to the sound of the poem. Internal rhymes link words and images, and lend the sonnet a greater sense of intimacy.
After considering form and genre, look at the poet's choice in language. Heaney focuses on the mundane language of everyday household chores - while we 'peeled potatoes' - but mixes this with a sense of joy and renewal in the 'gleaming' water and 'little pleasant splashes'. There is also a sense that this mundane image is a replacement for or a transposition of the religious: it happens 'When all the others were away at mass', and is remembered as a priest goes 'hammer and tongs at the prayers for the dying'.
Imagery plays a vital role in the poem. The description of the potato peel as 'solder weeping off the soldering iron' feels discordant and disconcerting, but it lends the moment of domestic intimacy a sense of industrial power that recurs in the sestet with the priest going 'hammer and tongs' at the prayers. Heaney is also deploying quasi-baptismal imagery in the reverent descriptions of the water - the 'bucket of clean water' - which makes the transposition of religious connotations complete.
By considering all of these aspects of the poem in turn during the planning phase of your essay, you can make sure you do not forget any vital element of it. When writing the essay, however, it would be a mistake to structure it based on individual elements of the poem. Rather, construct an argument based on the poem - for example, an argument about the movement of religious imagery in the sonnet - and use the technical details you have noticed to give body and evidence to your paragraphs.