There was no formal declaration. It was not the Ancient Greek custom to declare war on another state through an official envoy or document or ritual, like the Romans did. There are a few examples where a herald was sent to declare war on another state, but these are exceptional. We have more detail about the beginning of the Peloponnesian War than any other war from the Classical period, but there was no point at which the Spartans sent a herald or embassy to Athens to announce that they had decided to start a war.
Instead, what would happen when one state had a grievance with another was that envoys would be sent to demand redress or reparations. If the other state was not willing to accept the demands, the conflict might be submitted to arbitration by a neutral state. If the two sides were not prepared to submit to arbitration or if the process broke down, there might be a final mission to declare something along the lines of "give us X or prepare for war", and how the other state responded to that message would indicate whether or not they considered themselves at war.
If you're thinking "wow, there's a lot of room for ambiguity in this process," that is exactly the impression we get about the onset of the Peloponnesian War. A couple of major events in the late 430s BC persuaded the Spartans by 432 BC that the time had come for them to act; even so, there was no formal declaration, and they were officially only preparing for war, not actually commencing hostilities. There was still room for the two sides to negotiate some kind of settlement that would prevent war. The Spartans sent multiple embassies to Athens to make ever more straightforward demands, but each time the Athenian leader Perikles convinced the Athenians to dismiss them. Thucydides notes the state of tension:
Still, contacts continued in spite of [these events], and mutual communication. It was carried on without heralds [i.e. not on a war footing], but not without suspicion, as events were occurring which were equivalent to a breach of the treaty and matter for war.
-- Thuc. 1.146
For the historian, the moment when the war truly began was the point where "all communication between the two sides ceased, except through heralds" (2.1.1). This moment finally came in the early spring of 431 BC when the Thebans launched a surprise attack on the Athenian ally Plataiai, provoking Athens to war against Thebes, which proceeded to call on its ally Sparta for help. The Spartans responded by mustering the levy of the Peloponnesian League and gathering their army at Corinth. But even then, the Spartans thought there was still an opportunity to prevent all-out war:
Archidamos first sent off Melesippos, son of Diakritos, a Spartan, to Athens, in case she should be more inclined to submit on seeing the Peloponnesians actually on the march.
-- Thuc. 2.12.1
The implication is that the two sides were still not officially at war, and there was still a way for Athens to make the whole thing go away, even when the Spartan army was already mustering to invade Athenian territory. But Perikles could not be moved:
But the Athenians did not admit him into the city or to their assembly, since Perikles already carried a motion against admitting either herald or embassy from the Lakedaimonians after they had once marched out. The herald was accordingly sent away without an audience, and ordered to be beyond the frontier that same day; in future, if those who sent him had a proposition to make they must retire to their own territory before they dispatched embassies to Athens. An escort was sent with Melesippos to prevent his holding communication with any one.
When he reached the frontier and was just going to be dismissed, he departed with these words: "This day will be the beginning of great evils for the Greeks."
-- Thuc. 2.12.2-3
This final statement of the Spartan envoy is the real beginning of the war in Thucydides' account. It is not a formal declaration; it is not the announcement of a Spartan decision; it is merely the breakdown of a final attempt to resolve the conflict through words, after which both Athens and Sparta knew that weapons were next to speak.