The question in the title asks for the absolute beginnings of the modern domestic cat (species: felis cats). That is the way I read it.
It is believed that modern carnivores including the domestic cat evolved from miacids, a weasel-like creature from 60 million years ago. The first members of the modern cat family, pseudaelurus, evolved around 20 million years ago.
You can read more on the following page on domestic cat history.
P.S. You’ll have to believe in the evolution of the species and darwinism to agree to the contents of this post.