While the entirely fictional "Diary of a Wimpy Kid" is not totally applicable to our historical fiction journal writing activity, there are a few parallels. Author Jeff Kinney is assuming the identity of teenager Greg Heffley, just as you are assuming the identify of an Ancient Greek. Kinney tells his narrative in first-person and uses dates to move the narrative along, as will you.
Looking for actual examples of fictionalized historical diary entries and letters? Look no further. Read and listen to a fictionalized diary entry of a Native American boy living during the 1600's here. Next, read and listen to a fictionalized letter written by a pilgrim girl here. Notice that while the characters are fictional, the historical backdrop of their lives is real, as are the descriptions of their daily lives. The characters also sound like real people that may have actually lived. You are going to want to try to accomplish something similar.
Next, we will review a fictionalized account of an actual historical event known as the "Black Death" written by a modern teenager. Explains the teenage writer: "The Black Death was one of the deadliest plagues in history, and is said to have killed 30-60% of Europe's population in the 14th century. For my history class, we had to write journal entries from the point of view of one who died from this plague, including various, true details from the real event." You can check out her assignment "The Black Death: A Journal of One Who Died" here.
Finally, check out a few actual examples of real-life (not fictionalized) historical diaries and journals online here. Examining how real people from history wrote about their own lives may give you a few ideas about how to write about the life of your fictionalized Ancient Greek self.