This poem is acrostic, or a poem written whose lines come from a word. In this case, the word is Avada Kedavra. This poem also happens to rhyme.
A weapon of death is this spell,
Very useful to such as me.
A Dark Mark is enough to impel
Desperate cries to my decree.
A pest or a nuisance I can quickly quell.
Killing has become natural to me.
Ever ready to repel,
Distaste for society may be the key.
At death's door, my laughter is heard.
Voldemort has a mistress of death.
Relations leave me undeterred;
All may see me with their last breath.