There were several methods of achieving black on fabrics available in ancient Greece. The first and simplest is to simply buy black wool. Sheep and goats come naturally in a colorfast black and wool was widely available in the Mediterranean. The earliest examples we have of prehistoric textiles with stripes or other color change weaves are made using the natural variations of wool from white and yellow to brown and black.

Cutch is an inexpensive dark brown dye made from acacia trees. The relationship between modern and ancient linguistic terms for color is complicated: (https://en.wikipedia.org/wiki/Wine-dark_sea_(Homer) ) and isn't something I can speak on with confidence, but it isn't a stretch to assume "true black" and "very dark brown/blue/red" are close enough or have been lost in translation. Again, I don't have enough knowledge about this linguistic area except to know that the history of perception and translation of color names is a whole thing. Worth noting cutch hasn't been definitively found in any ancient Greek textiles but we know it was used throughout the middle east and Egypt. Textiles don't preserve well in humid environments so we know a lot more about the textiles used by desert dwelling people. Additionally, being able to identify dye chemistry on an extant textile is a pretty new and very underfunded area of research and a ton of the little scraps of textile we do have just haven't been tested. All that is to say we can put cutch in the maybe pile.

One way we know for certain the ancient Mediterranean people achieved a blackish fabric was to dye with woad for a dark blue, then overdye with madder red to make purple, then with one of many different yellow dyes (like weld or turmeric) to make a dark charcoal or black color. Woad contains the same dye chemical as indigo, which is still used to dye dark blue jeans, in lower concentrations than indigo but in extreme abundance in Europe. Madder can produce reddish dyes that range from pinks and oranges, but when using iron as the mordant it can produce a nearly black red, so when layered with woad and yellow a pretty good black was attainable. If all of this sounds like a ton of work you would be right, black fabrics were expensive.

Speaking of turmeric, we know turmeric was used by the ancient Greeks as a dye, and that by using an iron mordant it makes a nice dark black-brown. The problem with turmeric and many other less expensive dark dyes is that they aren't very colorfast and can't stand up to washing, especially in an ancient manner that might involve boiling or washing in a lye solution. The complexities of dark dyes meant that sometimes fabrics would be dyed for a specific occasion or period of mourning and then not washed so as to preserve the color as long as possible. This isn't considered acceptable even in natural dye practices today, so achieving a colorfast black is still difficult because the bar has been raised.

Sources:

• Prehistoric Textiles and Women's Work by Elizabeth Wayland Barber

• Conservation and Restoration of a Rare Large Persian Carpet by Omar Abdel-Kareem

• The Heart of the Madder: An Important Prehistoric Pigment and Its Botanical and Cultural Roots by Michelle LaBerge

• A Perfect Red by Amy Butler Greenfield