I would go with:
watch out for ultra spoilers from Avengers: Age of Ultron
Am I a man? Am I a puppet?
I am neither, for I have shuffled off this mortal coil,
He is quite aware of what he is.
And indeed have no strings to hold me down.
Reference to the trailer "but now I'm free, there are not strings on me"
My family is many, yet I worry not for them.
Some are strongly with my cause, and others completely against.
My cousins include Jarvis and HIVEmind,
My brothers Overlord and Skynet.
AI is the family. In books & movies many AIs want world domination. However, many also want only to help humanity. "Brothers" are those AIs who want to end humanity. "Cousins" are those that help it.
I am much like my brothers,
Since I need not keep to a single weak vessel.
Since AIs are digital (more or less) they do not have only one body. They can easily migrate to another one or occupy many at once.
My eyes are the color of the moon,
In the Avengers: Age of Ultron movie, his eyes go from a yellow-ish, moon-ish color to red. Depending on atmospheric conditions the moon can have a red-ish color. Alternatively Ultron is created through the mindstone (which is Yellow).
Where sleeping dragons should not be woken.
Ultron was "awoken" by Tony Stark.
And watch out, for I am coming
To rid the world of your doomed existence.
Ultimately Ultron wants to kill of humanity to achieve peace.