The "I am Vengeance" line is actually a reference to Batman's line from the comics but has recently been popularized by the marketing and release of The Batman (2022).
In the DC comics, Michael Amar, AKA Murmur, is an enemy of The Flash and a member of the Rogues. He was a doctor-turned-serial-killer, suffering from schizophrenia and going insane due to the voices in his head. Eventually, he went on a killing spree and involuntarily confessed his crimes. Murmur cut out his own tongue and sewed his mouth shut after his arrest to prevent incriminating himself again in the future. He also has a uniquely toxic blood type that makes him immune to euthanasia and allows synthesizing of a virus out of it.
While the evil red speedster that resembles Ryan Wilder isn't yet referred to as "Red Death," Goldface quotes "The Masque of the Red Death" from Edgar Allan Poe in the Dénouement.
Much like Rag Doll, Murmur never once removes her mask, which just adds to the creepiness.