100% in written sources, 0% "orally." We have manuscripts preserved to us, via copies of copies of copies, going back to about the 9th century CE, and they themselves were copies of copies of copies of Roman era-copies, which were themselves copies of Hellenistic-era copies, which were themselves copies of Classical-era copies. This gets us back to Homer and Hesiod, which date the the 8th and 7th centuries BCE. Homer of course has the famous epics, and Hesiod gives us the important Works and Days and Theogony, which is an encyclopedic account of the Greek pantheon. You can read a translation here from the ancient Greek.

From there, we go forwards in time to find other sources describing or celebrating the gods. From the 7th to the 6th centuries BCE, we enter the lyric period, in which we get hymns to the gods (the Homeric hymns, confusingly named), mythological stories (Bacchylides, Pindar, e.g.), etc. From there we get into the 5th century and the dawn of the Athenian tragedies, which zoom in on the plights of Greek heroic characters like Agamemnon, Oedipus, Antigone, and Medea. We then enter the Hellenistic age, and works like Apollonios' Argonautica and poets like Theocritus.

Then we enter the Roman period, especially here zooming to the end of the 1st century BCE, when the Roman poet Ovid composed his massive Metamorphoses. In it he preserved and retold many myths which had become obscure in his day, thereby preserving them. Moving forward in time from there, in the Roman literary milieu were numerous scholars interested in compiling and describing Graeco-Roman myth, and this continued down to the end of antiquity. From there, the copying of books continued through the centuries.

100 percent orally, until they were written down, at which point people continued, no doubt, to tell the stories.

The written texts likely influenced the contemporary oral tradition, just as the oral tradition changed and affected what was written. Of course, with conversion and various other tectonic movements in culture over time, the stories faded or were changed into the subsequent oral traditions that echoed through time among the Mediterranean cultures, but the subsequent narratives drifted a long way from the Classics stories of mythology as recorded by the many great authors who were working much like more recent folklorists, attempting to record and preserve oral narratives and traditions.

For the process of how the written works were preserved over time, one can look to the answer by /u/Alkibiades415. That user's capable answer might be taken incorrectly to imply that the Classic myths had no period of existence in oral tradition - but of course they did, for generations during an undocumented past.