In one of my novels, one of my characters dyes her hair -- this is easy enough to imagine in the modern day, of course. We all know where to find hair dye today, and the process of applying it is pretty simple. In the ancient world, and going even further back to the Bronze Age, it requires a bit more research. And by a bit, I mean that I probably lost a whole day to the process.
For your benefit and mine, here's what I found!
The process that was used during antiquity to dye hair is mind-blowingly awesome in its simultaneous simplicity and complexity. You see, during antiquity, Greeks and Romans and Egyptians dyed their hair using nano-technology without even knowing it! Of course the downside is that it required lead oxide, which probably didn't do much for their health.
For my book, the problem was, how the heck would my character get her hands on lead oxide or lime? And that kind of dye process is absolutely permanent--which is, of course, what my character was going for, but not at all helpful to ME for later events, and Homeric Greece was certainly not Antiquity. I can believe that these kinds of techniques were known in Egypt, however, and in the east. Troy by all accounts seems to be very rich in these kinds of things-- a center for trade. But my character, working under the radar, wouldn't really have access to what was needed for this technique, even by trial and error.
Another common dye, Henna, would most likely have been beyond her reach because it required trade to acquire from the east, but it was certainly available if someone wanted to go red. And it might have even been available (by trade) even earlier, which leads us to...
There are a variety of pigments that were available to people in Mycenaean times. Umber and Ochre for browns, reds and yellows, Bone and Carbon blacks, for, well, black for certain. But could any of these pigments be made into dyes? It would certainly require some kind of solvent (and I'm totally wishing I had blond hair of my own that I could trim and try mixing dyes in my kitchen sink about now.) The only information I was able to find on making ochre based dyes involved soy milk as the bonding agent. Earliest records of soy milk do not stretch back to Mycenaean times, even in China. Cow's milk has a similar amount of protein to soy milk, but I'm not sure it has the same enzymes to allow the bonding-- or it might require the addition of an acid to activate them (like Vinegar or wine, I'd imagine, though I have no idea how the chemistry would all work out), or maybe egg would do. Either way, for hair dye, these pigments probably aren't an optimal choice.
So what is?
Walnuts, actually. Boiling the fruit of the walnut tree apparently makes a dye which will darken as it oxidizes. While information on the cultivation of walnuts CERTAINLY dates back to Classical times, the information for the bronze age is a lot sketchier. Walnuts have been found in Europe well before the Bronze Age, though there are no signs that it was necessarily cultivated before the the classical period (pdf).
However, Walnuts do play a role in Greek Myth (relating to Dionysus), and working in a Homeric setting, or under the guise of the random naturally occurring walnut tree, or perhaps a walnut tree growing into existence by divine will, you can certainly get away with their use. They would have been known, even if they weren't a staple, and one would not necessarily have had to depend on trade in order to find a tree in the woods. Just breaking open the green outer shell would reveal the aspect of the dye -- in fact I witnessed this first hand just the other day, along with the stained fingers which resulted -- and a smart character could plausibly recognize its utility without requiring much of anything else.