The Old Testament indicates that Ezekiel knew of silk:
"I clothed thee also with broidered work, and shod thee with badgers' skin, and I girded thee about with fine linen, and I covered thee with silk."Julius Caesar had silk curtains, but we don't know for sure when trade between the silk-producing Chinese and the Romans began. By the end of the first century A.D., trade had been established, passing through the hands of a middle man, the Parthians or Persians.
King James Ezekiel 16.10
The Byzantine historian Procopius tells the story of how the Roman Empire acquired its own source of silk in book 8.17.1-8 of Procopius' History of the Wars. Although Justinian may have wanted to save the cost of importing the expensive fabric, the passage from Procopius first mentions not cost, but the desire to avoid trade with the enemy, the Persians.
The story Procopius tells is that some (Nestorian) monks from Sogdiana, who knew Justinian didn't want to buy silk from the Persians, told him that they could arrange it so that he would no longer have to buy from any other nation. They told him that it was impossible to bring the silkworms themselves from "Serinda" (an area north of India, i.e., China), but it was possible to take their offspring, cocoons, buried in dung. After Justinian promised to reward the monks well, they did as they had described, and brought the Chinese silkworms to the Roman Empire where the newly-hatched worms fed on mulberry leaves.
Source for trade on the silk road:
"The Silk Trade between China and the Roman Empire at Its Height, 'Circa' A. D. 90-130," by J. Thorley. Greece & Rome, 2nd Ser., Vol. 18, No. 1. (Apr., 1971), pp. 71-80.