When did humans begin to become self-aware and what did that process look like?

Julian Jaynes has a book called "The Origin of Consciousness in the Breakdown of the Bicameral Mind". It's a complex theory but focuses on language as the source for our supposed "self consciousness".

Burials probably came about when the invention of "personal names" came about. And then the Big Bang of consciousness Jaynes suggests is the invention of "I" as a metaphor for your imagined internal "Self" which consequently gives rise to an "Ego".

Read "Amos", a very early biblical work and then read "Ecclesiastes".

Old writings seem to lack subjectivity and the writer refers to themselves as their name and subsequently lack "introspection". But read "Ecclesiastes" and you get a fully introspective and subjective work of literature that reflects our modern self consciousness.

Look at my submitted post I did a week ago on Jaynes book.