I've been reading a lot lately about Sumerian civilization as well as other very ancient peoples. I recently realized I knew barely anything about the actual people. Of what I read The Epic of Gilgamesh came the closest, but it's mostly archaeology.
If the answer to my question is "not a lot", I'd be interested to know when in history we start to get some sort of idea about personality and psychology.
That probably has as much to do with your class and your reading; many older texts have been content to treat Mesopotamia as culturally and intellectually monolithic and been biased to regard individuality as an idea or ideal as unique to western societies. More recent scholarship however has stressed the understanding of Mesopotamian ideas of thought, personality, or psychology; a good overview of this is in:
Foster, Benjamin, "The Person in Mesopotamian Thought", in Radner, Karen, and Robson, Emily, The Oxford Handbook of Cuneiform Culture(New York: Oxford University Press, 2011).
See also Stephen Huston, David Stuart, and Karl Taube, The Memory of Bones: Body, Being and Experience Among the Classic Maya for a similar study of what personhood and personality meant in Mayan culture.
EDIT: Of course, feel free to inquire further if you have more specific questions in mind.
Perhaps this might offer a glimpse. Julian James - The Origin of Consciousness in the Breakdown of the Bicameral Mind. A fascinating read. http://www.amazon.com/gp/aw/d/0618057072?pc_redir=1406095814&robot_redir=1