According to William R. Cook, there is only one ancient account of slavery written by a slave that still survives: a letter written by Saint Patrick. We have many ancient documents that were written by slaves, but not documents about their experience of being a slave.
Patrick was born in Britain. He was kidnapped at age 16 and became a slave in Ireland. He served as a slave for six years before escaping and returning to Britain. Later he returned to Ireland as a missionary. Although there are many legends surrounding Patrick, historians generally agree that his autobiographical letter, now known as the Confession of St. Patrick, is authentic.
I was surprised to hear that there are no other extant autobiographies of slaves since there were many literate slaves in antiquity. Obviously slaves were not given the liberty to write about whatever they pleased, and slave owners would be unlikely to request candid biographies of their chattel. Still, I imagine some slaves wrote autobiographies, perhaps secretly. But it makes sense that such documents would not likely be preserved.
The lack of first-hand accounts of slavery may contribute to our rosy mental image of classical history. When we think of ancient Greece, we think of Plato and Aristotle, not the anonymous slaves who made up perhaps 40% of the population of classical Athens.
[Update December 2014: The information above comes from a Teaching Company course by William Cook. The original link is dead, and I don’t remember now which of his courses it was from.]