How does eulogy become elegy in this poem?
As you read the poem you are immediately struck by the unusual nature of the discussion of one who has died. A couple of definitions will help to answer the question at hand.
"Eulogy" - A speech or writing that praises someone highly, typically someone who has just died. Literally "good words."
"Elegy" - A poem of serious reflection, typically a lament for the dead.
In this very dystopian work, Auden approaches the citizen who is its subject in a much more analytical way than a typical eulogy. The "Big Brother" type references the various government organizations and their reports on the citizen, creating a more "serious reflection" than the good words of praise that would be indicative of a true eulogy.
It all centers around expectations. That which you would assume would come in a writing about the recently deceased never does.