Romans during the first century BCE didn't really keep time in the same way we do. They would track the lunar cycle to tell which particular month of the year this was and used this to determine when a year ended; but their timekeeping did not count back to any particular year. In later renaissance texts, you might see dates labeled as ab urbe condita or A.U.C. This means "from founding of the City," roughly 753 BCE in our time. If a Roman wanted to give an important date, they would list the particular Consuls who were governing at the time. For example:

Caesar, De Bello Gallico 1.2

M. Messala, [et P.] M. Pisone consulibus regni cupiditate inductus coniurationem nobilitatis fecit.

With Messala and Piso reigning as Consuls, he incited by a lust for power, created a conspiracy among the nobles.

Now, this just tells us how the Roman dated months and years. For dates within a month like April 10th, the Romans had a system which was very distinct from ours. They would use three "reference dates," the Kalends, Nones, and Ides. These roughly corresponded with the first, seventh and 15th of the month. They would then count backward from these, so April 10th in Roman time would be notated as 5 days before the Ides of April. This can also be found in Caesar:

DBG 1.6

Is dies erat a. d. V. Kal. Apr. L.

This day was the fifth before the Kalends of April. [March 28th]

One last thing, July was called Quintilis, or "Fifth Month" before Caesar's death. In honor of his calendar reform, it was later changed to July.

TLDR; If Caesar wanted to tell you his birthday was July 13 100 BCE he would say that he was born "During the consulship of Gaius Marius and Valerius Flaccus two days before the Ides of Quintilis.

Sources:

Caesar, Julius. De Bello Gallico

Cambridge Latin Course

A brief addition to the very accurate comment of u/_paul_132 - the Romans didn't use the Julian calendar before Caesar, he implemented it and in doing so changed the year to last 365 days (with an extra in February every four years). Before that, Roman years were somewhere between 340-355 days. The priests responsible for maintaining the calendars weren't fastidious about applying these changes every year, though, so Augustus had to make some more adjustments in his rule. (Suetonius and the Res Gestae cover these changes).

The names and orders of the months were messed with a fair bit by the emperors, too, changing them as honourifics (to themselves or to predecessors).