The "Destroying Angel" is Azrael's "shadow" self that she begins to encounter in her dreams after her experiences at the Well. She is the manifestation of the destructive half of her lineage, and she seeks to make her fall to Chaos.
(In some timelines, she succeeds.)
The "Destroying Angel" is constantly burning from within; her joints glow red-hot, soot covers her armor, and flakes of ash fall from her wings. Molten "tears" pour in an endless stream from her optics - the physical expression of the rage and grief that both destroys and sustains her.