There are a couple of myths that we can plausibly link to the Bronze Age Collapse and its effects on ancient Greece. First is the Trojan War, particularly what happened after the war was over. First we have the idea that maybe it was the Myceneans who were the invaders coming over the sea that we see in Egyptian texts. I consider it unlikely if only for the fact that the Myceneans were hit perhaps harder than anyone else by the Collapse. It makes no sense that a society so successful in war as to cripple multiple great empires would go into a precipitous decline.

Next is the memory of the return from Troy. If you recall the Odyssey, you hear what happened when many people returned from the war. The most famous is Agamemnon who was killed by his wife and her lover. Odysseus himself had to attack scores of men who threatened his claim to power on his return to Ithaca. Some people consider these stories a vague memory of a warlike period at the end of the Mycenean Era. This is a bit risky because there is no sense that these conflicts were cataclysmic and the stories have thematic importance that may distort their portrayal. Homer wanted to give counter examples to Odysseus's return (Helen and Menelaus are the best case, Clytemnestra and Agamemnon the worst case).

Another myth that might explain what went on in the Dark Ages is the Heraclids. The sons of Herakles wandered around and are considered the founders of most Dorian cities in Greece. They are probably a memory of the Dorian Invasion which we can chart by looking at the dialects in which ancient Greeks wrote (Dorians have a few linguistic quirks that distinguish them from Greeks in Attica or the Aegean coasts).

The Greeks definitely had a sense of a lost age. They lived among Mycenean ruins they could not explain. They figured that places like Mycenae were built by cyclopes. However, they had no knowledge of how the Myceneans lived nor did they have many links with them, so its hard to imagine they had a sense there had been a comparatively sophisticated version of themselves in the distant past.

There are a number of ancient sources that comment either explicitly on this period or are thought to be related to it.

Herodotus briefly alludes to the Trojan war, and to early Pelasgian inhabitants of Greece, but mostly mainly concerns himself with events from the 600s BCE onwards.

Thucydides has a much more systematic but brief summary of the early history of Greek communities and presents the legendary figures familiar to us from the dramas and Homer matter-of-factly as straightforward history - however, he does not seem to have a sense of a major break with the Mycenaean cultures, but rather has a narrative of continuity and development from what we know as the 12th century BCE to his own time.

A much more universal approach was taken by Diodorus Siculus, which I highly recommend reading. (http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Diodorus_Siculus/home.html) He, like Thucydides, presents what we consider to be mostly legends as straight-forward fact. Again, he doesn't seem to have a sense of a major break with the past, or that the Greeks of the classical age were anything but the direct cultural as well as genetic heirs of the great names of the past.

There are non-Greek sources available to us as well, such as Hittite and Egyptian letters that probably refer directly to events that were part of the whole Bronze Age Collapse, but there is little evidence that later Greek historians were aware of these at all.

Finally, you could read 1177 BC: The Year Civilization Collapsed by Eric Cline. It's not particularly technical or deep, but it will give you a short overview of the period.

So on the whole, I'm not sure that I agree that the Greeks had a conception of a the widespread collapse that happened ca. 600 years before the Persian wars, or that writing had been reintroduced in the 8th century BCE. I think that, to the extent that they thought about it at all, that they were the direct heirs of what we call the Bronze Age Mycenaean culture, that they spoke the same language, worshipped more or less the same gods and followed more or less the same cultural practices, and with a few exceptions (like Mycenae) living in the same places with the same names.

Nothing at all, because they didn't know about it. There were no Greek written records of any kind between ca. 1200 BCE until at the very earliest the 8th century BCE (it may be that parts of the Olympic victor lists were actually written down then, though that's unlikely; if not, then the date only shifts later). No classical-era Greeks were able to read any records from the earlier period (even supposing they were known).

That leaves the possibility of myth. That's always entertaining to speculate about, but of course it's only speculation. Here's what classical-era Greeks actually say about the period in between the legendary Trojan War and their own times. Thucydides 1.10:

It is reasonable to think that that Trojan expedition was greater than all in previous history, but still short of the modern scale. ... [T]he numbers embarked [in the Trojan War] do not seem very great for a combined expedition from the whole of Greece.

In 1.12 he goes on to describe turmoil caused by migrations (which are of extremely doubtful historicity), and at 1.13 he reaches his conclusion about the "Dark Age":

As Greece became more powerful, and the accumulation of wealth exceeded previous levels, the growth of revenues led in most cities to the establishment of tyrannies in place of the earlier hereditary kingships ...

So Thucydides is in no doubt that the history of pre-classical Greece is one of continually increasing population, wealth, and scale. He is not aware of any collapse.

Herodotos (writing a couple of decades earlier) has less to say, but at the end of his much briefer "archaeology" he does say this at 1.5:

I shall carry on with my story, describing both small and great cities of men in the same way. For those that were great in antiquity have mostly become small; those that were still great in my lifetime had previously been small. On the understanding that human prosperity never stays in the same place, I shall mention both in the same way.

So Herodotos takes a steady-state view of things. Some places get bigger and wealthier, others get smaller and poorer. The overall amount of prosperity in the world doesn't change (or he doesn't tell us it does).

So, Thucydides: Greece has been on an upward spiral since the "Trojan War". Herodotos: things pretty much stay the same. There is no awareness of a collapse. And there's absolutely no reason why we should expect any awareness of events that actually happened more than 400 years before any written records could possibly survive.