I know that there are quite a lot of Chinese sources talking about the Roman Empire and that there's archaeological evidence of trade (in the form of Roman coins and artefacts found in the Far East). I know nothing about the details, though.

Greek and Roman geographers had vague notions about a people they called Seres (basically meaning people of the silk) inhabiting the east. Pomponius Mela (I century A.D.) wrote:

Seres intersunt, genus plenum iustitiae, et commercio quod rebus in solitudine relictis absens peragit notissimum. (De Chorographia, 3.60)

The Seres come between [Scythians in the north and Indians in the south], a people of great integrity and very renowned for their trading, which they conduct while not being personally present, leaving their wares in the desert.

Pliny the Elder (same century) tells us more, and describes the Seres as a mild mannered people, but very insular and speaking a "harsh" language. He too talks about this kind of silent commerce (Naturalis Historia, VI.20.54). He also notes they're famous for their silk, even though he thinks it comes from trees (!). Later authors like Ammianus (IV century AD) repeats more or less the same stories, adding that they are very frugal (frugalissimi; Res Gestae, XXIII.6.68)

So people in the Roman Empire knew about the existence of the Chinese, but not much more. I don't know about any Roman mission to the east, official or otherwise, in what's left of Roman literature.