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The Didascalia Apostolorum, dated to AD 230, prescribe activities during Holy Week, including a fast on Friday:
Therefore you shall fast in the days of the Pascha from the tenth, which is the second day of the week; and you shall sustain yourselves with bread and salt and water only, at the ninth hour, until the fifth day of the week. But on the Friday and on the Sabbath fast wholly, and taste nothing. [v. 19] You shall come together and watch and keep vigil all the night with prayers and intercessions, and with reading of the Prophets, and with the Gospel and with Psalms, with fear and trembling and with earnest supplication, until the third hour in the night after the Sabbath; and then break your fasts. (v18–19)
Something similar is related in the Apostolic Constitutions (dated to 375–380) for Friday, the sixth day of the week:
But He commanded us to fast on the fourth and sixth days of the week; the former on account of His being betrayed, and the latter on account of His passion. But He appointed us to break our fast on the seventh day at the c**k-crowing, but to fast on the Sabbath-day. Not that the Sabbath-day is a day of fasting, being the rest from the creation, but because we ought to fast on this one Sabbath only, while on this day the Creator was under the earth. (5.3.15)
The Pilgrimage of Etheria (or Egeria), dated to a few years later, describes a more elaborate celebration, with five main stages:
The excerpts above are taken from the full text of The Pilgrimage of Etheria, which is too lengthy to quote here in full. The Encyclopedia of Ancient Christianity briefly summarizes:
Christians spent the time reflecting upon Jesus’ morning appearance before Pilate (Mk 15:2-15 and parallel texts) and the flagellation (Mk 15:16-20 and parallel texts). At noon, the wood of the cross found by Helena was shown to the faithful; then, for three hours, the people read the account of the passion, along with OT prophecies about Christ.
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I believe that the first instance of Good Friday (called Great and Holy Friday in the Orthodox Church) being recorded was in the diary of a Spanish nun named Egeria, written sometime between 381 and 385 AD, as summarized by Benedictine monk Dom Gregory Dix:
It begins on Passion Sunday with a procession to Bethany where the gospel of the raising of Lazarus is read. On the afternoon of Palm Sunday the whole church goes out to the Mount of Olives and returns in solemn procession to the city bearing branches of palm. There are evening visits to the Mount of Olives on each of the first three days of Holy Week, in commemoration of our Lord’s nightly withdrawal for the city during that week. On Maundy Thursday morning the eucharist is celebrated (for the only time in the year) in the chapel of the Cross, and not in the Martyrium; and all make their communion. In the evening after another eucharist the whole church keeps vigil at Constantine’s church of Eleona on the Mount of Olives, visiting Gethsemane after midnight and returning to the city in the morning for the reading of the gospel of the trial of Jesus. In the course of the morning of Good Friday all venerate the relics of the Cross, and then from noon to three p.m. all keep watch on the actual site of Golgotha (still left by Constantine’s architects open to the sky in the midst of a great colonnaded courtyard behind the Martyrium) with lections and prayers amid deep emotion. In the evening there is a final visit by the whole church to the Holy Sepulchre, where the gospel of the entombment is read. On Holy Saturday evening the paschal vigil still takes place much as in other churches, with its lections and prayers and baptisms….
There is some additional material on the history of Good Friday as well as the other days of Holy Week available from the Antiochian Orthodoox Christian Archdiocese web article, "The Historical Development of Holy Week Services In the Orthodox/Byzantine Rite"