I'd say that there are two main things I try to focus on and look for when reading poetry but they seem to go hand-in-hand. I mainly focus on detail of a poem, how in depth it can get - but with how much ease the poem can do it.
The second thing I focus on is how I can relate. How can I relate to this poem? It doesn't have to be something that's happened to me necessarily, but I remember a lot. I've remembered situations upon situations of stories of friends of families of etc. etc. etc. that I've thought about. It's a very big and interesting world, with so many things in it - so many situations. I don't have to suffer through a specific situation to feel their suffering, or in all lights - their happiness, gratitude, anxiousness, longing, and so forth.
I think these two go hand-in-hand, like I said, because I think if I'm looking for how I can relate to a poem then I will have to analyze the details of the poem itself.