The first civil calendar that explicitly used a 365.25 day year was the Julian calendar, instituted in 46 BCE. The Ptolemaic calendar was close, at 365 days, and the Egyptian year gained an extra quarter day immediately after the fall of the Ptolemies, around 30 BCE.

Prior to the adoption of solar calendars like these, most civilisations have used some form of lunar calendar. The Roman calendar prior to 46 BCE, for example, was nominally lunar, with a 355-day year corresponding to 12 lunar cycles of exactly 29.5 days each. So was the Hebrew calendar until the 3rd century CE.

Astronomical calendars in the Mediterranean world were more careful, because astronomers took it upon themselves to try to come up with unified 'lunisolar' systems, in which 29.5-day lunar cycles were reliably linked to the solar year. In lunisolar systems, the length of the solar year isn't measured in its own right, but calculated in terms of a number of lunar cycles. The idea was to come up with a Great Year that consisted of a round number of solar cycles, and a round number of lunar cycles (synodic months).

Different astronomers came up with different Great Years. The first influential one was Kleostratos' oktaeterís or '8 year period', corresponding to 99 lunar cycles = 2923.5 days. This produced a solar cycle of 365.4375 days. It was quickly superseded, but the oktaeterís remained influential well into the Roman era: Hippolytos of Rome was still using it as the basis for his calculations in the 3rd century CE.

Another very influential one was the Metonic cycle, measured by the 5th century BCE Athenian astronomer Meton. His Great Year as 19 solar cycles = 235 lunar cycles. This came out to a solar year of 365.2632 days. And the Athenian civil calendar was in fact remodelled based on this calculation, making it an early lunisolar calendar.

Also in the 5th century BCE, Harpalos came up with a 9 year cycle where 1 solar year came out to 365 13/24 days (365.5417 days). Oinopides of Chios used a cycle of 59 years = 730 months (1 year = exactly 365 days) as a starting point for calculating the solar year as 365 22/59 days (365.3729 days). A few decades later Philolaos, a Pythagorean from southern Italy, put the solar year at 365.5 days. Chaldaean astronomers, probably some time later, came up with a cycle of 222 months = 18.5 lunar years of 354 days each, called a saros. In some contexts this was refined to 223 months, which is also how saros is used in modern astronomy.

The decisive one was the year as measured by Kallippos of Kyzikos (4th cent. BCE). In the wake of Eudoxos' critique of the oktaeterís, he came up with a cycle of 76 years = 912 months + 28 intercalary months, producing a total of 27,759 days. And this works out to a rate of 1 solar year = 365.25 days.

The Julian and Alexandrian solar calendars, therefore, are based on the Kallippic cycle. (So I guess you could say that they're technically actually lunisolar. But no one will care.)

Tweaks continued to emerge later on, but 365.25 days remained the standard. Hipparchos and Ptolemy both determined that the figure was still slightly too high; they both measured it as having an error of 1 day every 300 years. As a result, their more refined figure for the solar year would be 365.2467 days. But the error was already so small that the 365.25 day figure was adequate for most purposes.

As a postscript, the true error was larger than Hipparchos and Ptolemy measured: the Kallippan 365.25-day calendar actually has an error of 1 day every 128 years. Steps were taken to address this in the 1500s CE, resulting in the modern Gregorian calendar, with a 365.2425-day year, which is so close to the correct figure that it will take more than 3000 years before it's off by 1 day.