They would be a mixture of family master painters and employed painters!
As for the question about Charms... While I imagine the charm to make the paintings move might be simple since it's simply using Transfiguration to give an inanimate object movement.
I imagine the Charms used on paintings of specific people such as the Hogwarts Headmasters, which are meant to act and behave like a specific person must be extraordinarily complex and probably require a mixture of Potions and Charms to work. Though maybe I've just been reading too much fanfic haha