The concept of genetic heritability is obviously a very new one, but there are some broad theories on the nature of human reproduction in antiquity that have survived.

In Greco-Roman thought (as well as the Hellenistic-inspired intellectual traditions of the Judeo-Islamic world and Western Europe), the woman was seen as a passive receptacle for the true “creative principle” of reproduction, which was believed to be exclusively a masculine contribution. For example, Aristotle believed:

“His general conception of “seed” and of the process of generation is in keeping with three other aspects of his thought: the Aristotelian theory of the four causes (the mother being only the material cause of generation while the father is both its formal and efficient cause); a general economy of interchangeable body fluids; and the prescientific assumption which Bachelard called the myth of the concentration of substance. The woman produces a seed that only serves as food for the embryo. The fashioning of the embryo is a form of coction that in a famous metaphor Aristotle compares to the curdling of milk.”

Meanwhile, a Hellenistic-era medical writer re-affirmed the notion that women have no influential part in reproduction:

“The seminal duct, which begins in the uterus, passes through each ovary and after following the flanks of the organ as far as the bladder, empties into the neck of the latter. Observations indicate that the female seed does not seem to be collected for the purpose of conception, since it is discharged externally; I have taken a position on these facts in my treatise On Seed (Diseases of Women 1.4.93-98).”

Even the famous Roman physician Galen, who remained definitive in the Islamic world and eventually Europe after translation from Arabic to Latin until the predominance of Avicenna, hosted a milder version of this view:

“In spite of his reverence for Aristotle, he has the idea that the woman produces a seed useful for generation; but it is nonetheless inferior to the male sperm, notably in terms of agility and heat. For him too, the creative principle resides in the male sperm, and when Galen reflects on the determination of the embryo’s sex, he adopts an explanation based on the criterion of sides: if the seed falls into the right side of the uterus it will give birth to a boy, but on the left a girl.”

But Galen still theorized that both men and women produced 'sperm' (or male seed and female seed), and although women have 'weaker (or more recessively expressed) sperm', both sexes contribute dualistically to the generation of the child. This is interesting, because it's reminiscent of dominant/recessive traits in the study of Mendelian genetics:

"In fact, both partners alike contain both male and female sperm (the male creature being stronger than the female must, of course, originate from a stronger sperm). Here is a further point: if both partners produce a stronger sperm, then a male is the result, whereas if both produce a weaker form, than a female is the result. . ."

This is why pre-modern notions of the “homonculus” (literally: little human) imagine a fully formed human already within the sperm— it wasn’t based on observation of the offspring’s traits, but the belief that the man, being the more active, vigorous participant in reproduction, is the primary/dominant sex in the origination of children.

So, at least in the West, the sperm of the father was seen as the substance that passed on his likeness and characteristics, while there are more confusing and ambiguous views on the role of women in defining her own children’s traits.

As far as the heritability of personality goes, some classical figures seemed to have believed that 'personality' is grounded in an individual's racial or ethnic nature -- for example, Aristotle claims the 'northern races' are more "spirited" than the 'southern races' (who are more "intelligent"), perhaps an allusion to Alexander the Great. Others, like Plato, however, seemed to believe in "nurture" over nature -- in his ideal of a utopia, he explains that children would have to be indoctrinated that they were made of "gold, silver or brass/iron" in order to efficiently structure a segregated yet functioning Spartan-like society, perhaps an indication that he believed personality wasn't necessarily inherited by one's birth status.

Sources:

https://journals.lib.unb.ca/index.php/flor/article/viewFile/19171/20845

https://journals.openedition.org/cliowgh/339

​ https://www.jstor.org/stable/4330958?seq=8#metadata_info_tab_contents

Edit: Formatting Edit 2: Expanding on Galen's theory Edit 3: Wording

So, I don't know the classical debates, but as late as the 18th century most debates about reproduction and inheritance in the western European context consisted of debates between two competing theories -- preformationism and epigeneticism. The preformationists (which was called, by contemporaries, evolutionism -- not the same kind of evolution as Darwin) essentially believed in the Matryoshka doll model of reproduction: that either the male or female of every species contained eggs or sperm which contained within them the complete organism of the next generation in miniature (homunculus). That miniature organism was thought to contain its own eggs/sperm containing their offspring in miniature, and so forth. The thought was that when 'fertilization' (for lack of a better word) occurred, the compressed organism in the germ cell would begin to expand and grow. These people thus explained resemblance and heredity in this way (that's why you look like your father and his father and his father's father, etc.), and tried to explain divergence by arguing that the non-homunculus parent 'contributed' in some way to the child's development during conception and gestation. With respect to inheritance of temperament, you can read something like Laurence Sterne's novel Tristram Shandy, where disruptions in the offspring because of mishaps during the moment of insemination are a major plot point.

There are complex reasons for what made this apparently implausible theory seem plausible for a long time, mostly related to difficulties in explaining something like the formation of complex structures from very simple cells. (They would try to look at embryos in different stages of development to see if they were 'unfolding' as postulated or self-organizing.) Although we would think that the epigeneticists were obviously the precursors of modern scientists in their belief in organisms forming during gestation, this is not exactly the case: Without genetics, the epigeneticists were forced to postulate strange and hard to study 'life forces' responsible for giving organisms their form, and this seemed to many like substituting one impossible to explain phenomenon for another.

Pushback on preformationism started to come in part through examples of hybrid or animal cross-breeding and human inheritance. One of the more famous treatises is Maupertuis' Physical Venus (1745), which tries to find a middle way between these two theories. Maupertuis' argument was occasioned by the carnivalistic display of an albino African, and he discovered in part that albinism showed strong hereditary tendencies, but that it could 'cross' from a maternal line to a paternal line. This made it very hard to defend a preformationist belief that future generations were always contained in either the mother or the father (i.e., that it is either women or men that bear all the homunculi, and the other sex merely contributes some sort of 'activator').

Still, these debates went on for a while, and were still 'live' questions at the time Kant was writing in the late 18th century. Kant has elaborate arguments for how an initial 'breeding pair' of humans (or several) contained 'dispositions' for different skin colors / races, and as early humans spread to different parts of the globe, the skin color best suited for a given climate became the norm, and this original 'plasticity' of dispositions solidified and is now fixed in the form of distinct races.

The evolutionary biologist Stephen Jay Gould has a very interesting and accessible essay on just this question entitled "For Want of a Metaphor", in his book The Flamingo's Smile.

One thing I think is worth pointing out: prior to the 19th century, the main question of learned people in the West who studied this topic wasn't "how does heredity work?," as we would phrase it, but "how does development work?" In other words, how do you go from an acorn to a tree, an egg to an organism? That's what Aristotle cared about, that's even what Darwin thought the question of heredity was really about. That's the amazing and impossible biological phenomena to be explained, how something huge and complicated comes out of something so small, undifferentiated, and essentially "simple" looking.

So this is why all of the older discussions are about things like "is there a tiny person inside your sperm" which, aside from looking quite silly to us today, seem to miss really obvious questions like, "why does heredity sometimes look like a blend of parents and sometimes not?" (e.g., the problem of the occasional black sheep, or "sport") It's also why early discussions of heredity look confused to a modern eye: because they're not about heredity as we conceive it.

When did this switch from development to modern notions of heredity (e.g. traits, somatic vs. genetic, and so on, much less the post-Watson-Crick conception of it as a "code" problem) take place? Basically in the late 19th century, around the time Darwin was alive. Darwin's own writings on heredity were still preoccupied by the development problem, though he was starting (not very successfully) to edge into the modern heredity problem. His followers — Huxley, Galton, and others — were the ones who saw that actually the heredity problem was more important to his theory than the development problem (which is still interesting, but isn't the crux of evolutionary theory), and started reforming it. Galton's influence in particular is extremely evident in our modern conception of the issue (he coined the nature vs. nurture problem, he developed the statistical methods for looking at it, he also extrapolated it into eugenics, all for better and worse). This is also around the same time that Mendel was working on his own approach to heredity.

When you see historical smart people being so obviously dense (as is a common response to Aristotle's views of nature, which often seem so obviously wrong), the insight to have is not, "gosh, we're so much smarter now," but, "oh, they really were asking different questions than we are today," which is usually correct. (Aristotle makes a lot of sense from within Aristotle's own worldview, he makes almost no sense within a modern worldview. So the trick to understanding Aristotle is to reconstruct his worldview.)

Pre-modern observers, especially agriculturalists, understood what we sometimes call "folk heredity." This is what people can see if they spend time working with animals, large families, and so on: offspring usually do represent parent stock, but not perfectly. There are sometimes very dramatically inherited traits (Habsburg chins, etc.), and there are also sometimes "sports" (rare one-offs that seem to resemble neither parents; today we would say these are the result of double-recessives in most cases). To go much further than that appears to have been rare except in the medical synthesizers like Aristotle (whose theories make not a lot of sense, but when did that ever stop him?) for quite a long time. To be fair: a good understanding of heredity requires quite a few other ideas to come into place (to imagine that we are made up of trillions of tiny cells that are all working from a common "codebook" of proteins that it uses to direct the development of the entire organisms is kind of an unobvious thing, much less the complexities of Mendelian ratios, meiosis, and so on), and we've been raised in societies where discussing heredity has become commonplace and core to our identities ("he gets it from his mother"/"I have a genetic risk factor for X"/"I'm X% this and X% that, so you know how I am").

For a really readable discussion of the development of modern genetics, and some of what came before it, Peter Bowler's The Mendelian Revolution is great.