For me, poems tend to be different from other forms of writing in two ways: the form of the poem and the use of language in the poem. Therefore, when writing about or interpreting poetry, I always try to begin by considering those two things.
Under form, it might be important to recognize that a poem is a sonnet or is written in open form, for example; the form of the poem can often have a direction connection to the poem's meaning. The open form of Walt Whitman's poetry matches the content very well; the poems seem inclusive, spontaneous, liberated, etc. The sonnet form always seems, to me, to be much the opposite: formal, polished, refined, practiced, etc.
Under use of language, it might be important to identify and reflect on an image that is central to the poem. Other elements to consider might include uexpected ways that the poem has of saying something, associations that you as a reader have when you read particular words in the poem, etc.
If you can talk about both form and language in poetry -- and, even better, make a meaningful connection between the two (do they work with or against each other?) -- I suspect that what you write will be good.
There are other items that you might consider, of course. The link below will take you to one of many online resources that walk through a set of questions that you can apply when preparing to interpret a poem.
As a final note, I would recommend focusing solely on the poem and not making any references to the poet. The poet's thoughts about the poem are, as famously phrased by the New Critics, both unavailable and undesirable.
When writing about poetry, you want to discuss the structure, literary/poetic elements and literary/poetic, techniques. Firstly, in structure, along with identifying the genre and sub-genre, e.g., lyric and sonnet, you will want to analyze the rhythm and the meter, which when added together create the metric pattern, e.g., trochaic tetrameter. You will also identify the rhyme scheme; e.g., abab cddc. You will also discuss the structure of stanzas and individual verses [verse(s) is the poetic term for line(s)]. These considerations also include blank verse and free verse and other variations on rhythm, meter and rhyme.
Secondly, you will analyze and discuss the literary device of literary elements, although when applied to poetry these are called poetic devices and poetic elements. Some poetic elements are theme; speaker/narrator: i.e., identify whose is the poetic voice; tone; mood, synonymously called atmosphere; and the metaphoric basis of the poem. In poetry, there may or may not be something like a plot. A narrative poem or a dramatic monologue as developed by Robert Browning might have some kind of a plot structure. However, poetry does have the progression of ideas and many kinds of poetry have a diametric turn in the topic, such as sonnets do.
Thirdly, you will identify and analyze the poetic device of poetic techniques. Poetic techniques differ from poetic elements, though both are poetic devices, in that they are optional additions used at the discretion and choice of the poet. For example, aside from the prevailing poetic metaphor employed in the construction of the poem, the use of metaphor and simile is up to the poet's choice. Some poetic techniques are similes, personification, metonymy, irony, sarcasm (distinct from irony by tone and intent), diction, and synecdoche.
As to interpreting a poem, each of the points discussed above, structure, elements and techniques, will reveal the meaning of the poem to you. Once you see the component parts from theme to structure (e.g., underlying metaphor; diametrical turn in topic) to diction to similes and all the rest, you will begin to see the deeper aspects of meaning through the clarification of ideas, the juxtaposition of ideas, the multifaceted meaning of ideas, the expansion of ideas, etc. Remember that structure plays a big part in facilitating the interpretation of poetry, similar to the importance of structure in fiction, as per one example, when the story is structured in a frame.