Although the exact date is unknown, an estimation can be made. Isaiah was a prophet to 4 successive kings of Judah: Uzziah, Jotham, Ahaz, and Hezekiah.
Isa 1:1 The vision of Isaiah the son of Amoz, which he saw concerning Judah and Jerusalem in the days of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah.
Uzziah reigned for 52 years from 790-739BC, but he was struck with leprousy by God as punishment for trying to usurp the duties of the priests unto himself. This meant he was not allowed public contact, and his son Jotham therefore had to become king. Jotham ruled 750-731BC, so the latest possible year Isaiah could have been a prophet to Uzziah before the co-regency with Jotham is 749BC. Tradition says that Isaiah was cut in two with a wooden saw while he was hiding in a tree-trunk by troops of King Manasseh (695-642BC), so the earliest possible date for his death is 695BC. This makes the minimum time period possible for Isaiah's prophetic life 749-695BC, or 44 years. It is safe to assume he was 30 when he started, because (many years later) the Pharisees would not allow anyone to be a rabbi or a member of the Sanhedrin until they were 30. This would make him a minimum of 74 years old when he died, which is a credible age, and would make the latest Isaiah could be born around about 770BC. (Since the actual date is unknown all this is based on assumptions, but you have to start somewhere).
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The prophet Isaiah's time can be fixed by reference to the reigns of the Kings in which he prophesied.
The vision of Isaiah the son of Amoz, which he saw concerning Judah and Jerusalem in the days of Uzziah, Jotham, Ahaz, and Hezekiah, kings of Judah.
Thus the first period of his ministry was during the reigns of Uzziah 792-740 BC and Jotham 750-738 (as regent) and 738-732 as ruler. Isaiah's second period of work began in the reign of Ahaz from 735 - 715 and continued into the fifteenth year of Hezekiah's rule, after which he retired from public ministry.
Isaiah was a prophet during the reigns of 4 kings of Judah (739-686BC): he died about 681BC when he was cut in two with a wooden saw under King Manasseh [from the 'Study Bible' by John MacArthur], so it can't be any later than that!
The prophet Isaiah lived in the 8th Century BC in the Kingdom of Judah.
He completed the writing the book of Isaiah around 732 BC.
See related links for more information on Isaiah and the Book of Isaiah.
In Isaiah 1:1, Isaiah introduces himself as the son of Amoz and says that he is writing about events in Judah and Jerusalem during the reigns of Kings Uzziah (Azariah), Jotham, Ahaz and Hezekiah. Of course, he does not say his witness includes the entire reign of Uzziah, but even the reigns of just Jotham, Ahaz and Hezekiah spanned 55 years between 768 and 686 BCE, and Isaiah was a prophet for at least 40 of those years.
Isaiah's prophecies finish in chapter 39 of the Book of Isaiah, and are followed by the work of an anonymous prophet now known as Second Isaiah, who lived during the Babylonian Exile.
Isaiah as a prophet occurs during the years 649 BC to 584 BC, according to the Chart 50 VII (6), page 68 Appendix 50, of the Companion Bible, by EW Bullenger