My way of praying with poetry is not really unique. I do lectio divina this way, also. This is what I do:
I look to see if I'm drawn into the poem.
What is especially striking?
Does it help me understand God better, or the world, or myself?
What mood is the poem and what mood does it draw me into?
What are the metaphors? What do they say to me?
What is the tone?
Is the voice of the poem helping me identify with the poet, or what he is saying?
Do I really understand what the poem is saying?
Poetry is the opening and closing of a door, leaving those who look through to guess what is seen during a moment. ---Carl Sandburg