I apologise this is nothing like a full answer. However 'The Periplus of the Erythrean Sea'[which roughly means, the Indian Ocean] written in ancient Greek but during Roman times, most likely 1st Century AD, translation available by internet search, is a practical account of trading ports and routes known to Greco-Roman merchants, what was bought and sold there, dangers en route etc.
This contains quite a lot of information about trading to India and a little about Burma, but says no ships are known to have sailed further. It mentions 'a city called Thina' somewhere further inland from which silk is brought by traders. This is probably a garbled reference to China.
The Romans did have silk as a luxury good, so must have had indirect trade with China as the only producer at that time, through 'middlemen'or a series of middlemen. However, the Romans were often in conflict with the Parthians and Persians, whose empires tended to be in the way of direct land travel from the Mediterranean area to China.
I am sure I have read of other, clearer references to China by ancient Greek or Roman authors, but do not recall which ones. My recollection is that they knew it was there and was a large, civilized country, but had little other hard information.