My first response is that writing in ancient times, even letters, was always intended for an audience of more than a few people. The only writing I can think of that approaches journal writing with references to private topics: moods, culpability, remorse, self-questioning, trivia of daily life, is the Confessions of St. Augustine, on the eve of the medieval period. I'm sure others will have sources for you.
"Dear Diary,
It's nine o'clock, the children are fast asleep, and I'm going over these latest accounts from Trajan's Market before I hit the sack." Ancient people didn't conceive of their world or of themselves in this way.
I have a character who wants to journal, but I strive for time period authenticity. I think that the desire to reflect in the written word must have always existed. Perhaps it was done in a way unfamiliar to the modern eye, or just never has been discovered, which won't help my research. I imagine my heroine could write letters to someone close to her, like a confessor friend. I also wonder if she was to write letters to herself or a close friend, would it best be done on pieces of papyrus, parchment or in a codex? Any thoughts??