You'll find that diaries are much less common before the modern period than you'd expect. There's a sudden spike of published diaries in the early modern period, but they're relatively rare before. They're not totally unknown in antiquity, but they take a form very different from what we're used to, with daily catalogs of whatever nonsense happened that day. The Ephemerides are sometimes called the "Royal Diaries"--they were a journal record of sorts kept of Alexander's activities by his court secretary Eumenes. But they're not what we would consider a journal and what we have of them, a few quotations, seem sort of spurious--they're mostly about Alexander's drinking bouts and his final sickness. As the OCD says, "Eumenes no doubt used his name and position to inspire credence, but there is no evidence that he represented the diaries as archival material," and in any case they're not really what we're talking about. Herodes Atticus, the famous sophist, supposedly published a collection of lectures and diaries, but these are almost certainly literary diaries, using the form for rhetorical and philosophical purposes (some suppose Marcus Aurelius' Τὰ εἰς ἑαυτόν to be a variation on this form). The elder Balbus, a Caesarian who edited Hirtius' works, published a diary that is now lost--there's not really any reason to suppose that this was not a literary publication either, though it too is lost. Aelius Aristides, during his period of incubation in the sanctuary of Asclepius, compiled a record of the revelations given to him by the god through dreams, which is sort of like a real diary, but not really--it is, however, important as the most complete collection of first-hand accounts of religious experience. But I hesitate to call it a diary or a mundane record. There are things like the Itinerarium Egeriae, a travel-log, or various Commentarii written by leaders and educated people. But neither of those are diaries--commentarii are really memoranda, which by the Augustan period were being compiled for literary purposes and should be thought of more as autobiographies than daily accounts (Caesar's commentaries are a little different than many, since we think, although we don't know, that they were mostly edited and compiled from his litteras ad senatum).

Probably the closest thing we can come to are letters. But not all letters published in antiquity were candid or unedited. Seneca's letters, for example, may not actually ever have been sent to anyone (I personally think that there was no Lucilius at all, and if there was these letters were never sent to him). If they were, they were certainly written for publication--their style is distinctly literary and polished, and they exhibit features more usual to a lecture than a correspondence. Cicero published an edited collection of letters during his lifetime, with the rest published after his death by Tiro and Atticus. They are of rather varying type--Shackleton Bailey noted that they were surely not written with publication in mind. Nevertheless, some of the correspondences were naturally quite different from others--Cicero obviously did not write to his brother Quintus the same way that he wrote to Caesar. The letters ad Familiares tend to be the most candid, as this collection was only published after Cicero's death and certainly Cicero never thought of publishing them--some are long and rather formal, but many are quite short little check-ups on people. Again, not a diary, but often rather candid.