Some ancient Greek thinkers believed that there were different kinds of blood in our bodies. There is the blood that is clearly red. But there is also black blood.

This mirrors the distinction between arterial and venous blood. Arterial blood is redder because it has been oxygenated.

Note that the Greeks did not know about the distinction between arteries and veins until Praxagoras of Cos at the end of the 4th century BC.They also did not understand the role of the heart in the circulatory system or where blood came from. Aristotle comments in the History of Animals that it seems plainly impossible to trace the courses of the veins; such are the difficulties of making sense of the circulatory system. But they had theories.

The Greeks generally were aware that blood is red. Plato tries to explain this in the Timaeus:

For the fire cuts the food and following the breath surges up within, fire and breath rising together and filling the veins by drawing up out of the belly and pouring into them the cut portions of the food ; and so the streams of food are kept flowing through the whole body in all animals. And fresh cuttings from kindred substances, whether the fruits of the earth or herb of the field, which God planted to be our daily food, acquire all sorts of colours by their inter-mixture ; but red is the most pervading of them, being created by the cutting action of fire and by the impression which it makes on a moist substance ; and hence the liquid which circulates in the body has a colour such as we have described. The liquid itself we call blood, which nourishes the flesh and the whole body, whence all parts are watered and empty places filled.

So, blood is red because it takes on that color from the fire that has acted on it.

Plato, like most other Greeks, believed that blood was cut-up food. Homer believed this too. That's why the gods don't bleed blood in Homeric texts. For instance, he writes: "and out gushed the immortal blood of the goddess, ichor, such as flow in the veins of the serene gods." The gods bleed ichor instead of blood because the gods don't eat food: they eat ambrosia.

But even Plato, who develops a theory to explain why blood is red, believes that he has to explain the presence of a black liquidy substance in the body.

He thinks that black bile is the source of many diseases in a living thing. What is black bile? It is "is only stale blood, and which from being flesh is dissolved again into blood." Black bile is actually just stale blood: this makes sense because blood, being cut-up food, is meant to replenish our flesh as it deteriorates over time. Black bile is blood that has returned to the bloodstream when it was meant to replenish needy parts of our body. That's why he says that it has "violated the laws of nature."

I'm happy to answer follow-up questions, but the short version is really quite simple: the Greeks knew that some blood was black and other was red but they didn't have the anatomical resources to explain arterial vs venous blood. I doubt Homer would have maintained that all blood was red or black; he would say it is mixed. The reason why I doubt that he would have is that there is a remarkable continuity between the theories of blood that Greeks were working with until the end of the classical period. (I do not mean to deny, of course, that Plato, Aristotle, etc., disagreed with each other, but everyone -- such as Empedocles, the Hippocratic medical writers, Plato, Aristotle, etc. -- all agreed on the same phenomena that need to be explained, and Homer seems to share the same common ground.) The Greeks also lacked the physiological resources to explain the production of blood, and so they relied on a theory of circulation that combined the process of blood-making with the process of digestion, such that blood was merely cut-up food. They then need to explain why blood has a color different from the color of the food that it once was. This then ultimately culminates in an explanation of why some blood is red and some is black.

/u/voltimand is absolutely correct to point out that vocabulary, especially vocabulary for physiology, doesn't line up precisely between cultures and philosophical traditions (c.f. 足, simultaneously "foot" and "leg"). The Greeks identified physiological properties differently than we tend to today in contemporary English. But, since these sorts of questions about Homer specifically are very common around here I want to point out what I would hope is obvious, but apparently isn't.

The Homeric Poems are poetry.

Blake can describe tigers as burning bright (even using the adjective instead of the adverb), and no one bats an eye. But when Homer calls blood κελαινεφής everybody loses his mind. Why? The Homeric poet doesn't refer to the sea as "winey" because he thinks it's the same color as wine, any more than he thinks that Hera is literally "ox-faced." Any other poet would be understood as metaphorical. So why does Homer draw so much attention?

For starters, this isn't new. Homer occupies an unusual place in western thought, as the Homeric Poems are simultaneously identified as the genesis of "western" literature, and therefore remarkably sophisticated, while also being the product of a fairly primitive late Dark Age Greek culture. Homer, the unconscious reasoning goes, could not have understood complex metaphors or metonymy, because Homer didn't have the literary theory. And indeed, the famous Homeric Simile seems fairly unsophisticated and clunky, if enduringly beautiful, beside the understated brilliance of Eliot's "April is the cruelest month." Attempts to psychologize or physiologize the Homeric texts go way, way back. Most notorious on the internet is the silly blue thing, but probably best known to scholars and people who read Greek is the old conjecture that since gods only ever appear to single individuals and can't be seen in that moment by anyone else, the Homeric poet must be describing what are actually psychological conflicts entirely within the character's mind, but doesn't have the vocabulary or sophisticated understanding of psychology to be able to voice those ideas without attributing them to a god. Neither idea is taken especially seriously by Homeric scholars today.

Second, Homer's vocabulary for blood is much more sophisticated than simply "black." Sometimes he does indeed call blood αἷμα κελαινὸν, "black/dark blood," using the word κελαινός, "black, dark." So, at Il. 1.303 Achilles threatens that Agamemnon's αἷμα κελαινὸν will run on his spear. But elsewhere Homer uses different words. At Il. 4.140, for example, he calls blood κελαινεφὲς, from κελαινεφής, a compound word from κελαινός, "dark," and νέφος, "cloud." This is the same adjective used of Zeus when the poet wishes to emphasize Zeus' power and anger. Clearly there's something ill-omened and grave about the term, which one might translate somewhat prosaically as "black, clouded blood," which the poet is trying to emphasize by its usage. Further, Homer has several words for dark or black. If we return to Il. 1.303, the poet has Achilles refer to his νηῦς μέλαινα, his "black/dark ship" only 3 lines earlier at 1.300. Egad! How can this be?! The poet has two words for black, and he's used both of them within two lines of each other?! It gets even worse: Homer uses μέλας to describe blood as well, e.g. at 4.149, only 9 lines after he calls blood κελαινεφὲς. To muddle things even more, κελαινός is a Homeric term, not an ordinary Greek word. It's used pretty much only by Homer, the tragedians, and anyone trying to sound particularly epic. The ordinary Greek word for dark is μέλας. So already in his regular vocabulary the poet's choosing to use unusual words. Why? Because he's a poet. He's choosing his words for poetic effect, not because he's a prosaic physician.