When I was in college, I was given a list of paints we had to have in our palette for a painting class. Carbon black, bone black, burnt sienna and so on.
It never, ever occurred to me to think about the names or what the ingredients were and this information was never mentioned, ever. In all of my art classes, never once. It wasn't until recently that I realized that carbon black was originally made from, well carbon - the burnt charcoal of wood. Bone black was originally made from the charcoal of bones. Ivory black? You guessed it. Charcoal of elephant tusks.
Carbon black in particular is perhaps the easiest paint to make at home, no chemicals needed. Essentially, you take some wood, put it in a container in a hot fire until it turns to black charcoal. Now, you don't want the wood to actually start on fire. Ashes are not what we want. The wood needs to be heated without oxygen. Several layers of tin foil, a cast iron dutch oven, those cast iron camping sandwich presses for bonfires or a tin box will work.
Bone black is made in the exact same way except with bones. The artist manual Il Libro dell'Arte by Cennino Cennini from the early 1400s recommends, "the second joints and wings of fowls or of a capon" for bone black. So, chicken wings. I'm sure choosing a capon (which is a castrated rooster, by the way) adds that special something, but I'll have to do without.
So once it is cooled down (I start a fire in the fireplace, tuck the tin in there and just leave it overnight) you grind down the charcoal and there you have carbon black pigment. *Use a dust mask for this part.*
From there, you can follow one of the many recipes for paints or watercolors - I use Gum Arabic and honey but I don't add glycerin for using the watercolors on wood. You certainly can, but I don't find it necessary to extend the drying time.
You can see there are differences in the warmth and coolness of the homemade black paints.