Recently someone asked me if there were Jewish states as we consider states today before the Romans.
I guess the Jewish kingdoms BCE were not like the states today. In fact, even in Europe the modern-style state only emerged in the 18th-19th centuries. Unification of Italy, Unification of Germany, and even the formation of the United Kingdom.
Before that there were small kingdoms with some empires like the Babylonian, Persian empire, Greek Empire, and Roman Empire, and the Arab / Muslim Caliphate.
The Jews were mentioned in the history of all of these empires either as people who were conquested or people who were set free or where Jews could live as second-class citizens called Dhimmi who are required to pay an extra fee called jizya to be protected. (aka. "protection").
Biblical evidence
If you consider the Bible to be a divine text, then I'd guess that should be enough.
On the other hand if you don't consider it divine then it is basically a mix of anecdotes from the creation of the world followed by a history and law book.
Archeological evidence
There is plenty of archaeological evidence that proves the existence of the two Jewish kingdoms. Your are more than welcome to visit Israel and go on an archeology tour to see whatever you'd like to see.
In the meantime let me leave this quote here: "historians and archaeologists agree that Israel and Judah existed as separate kingdoms by c. 900 BCE and c. 850 BCE, respectively." from the History of Ancient Israel and Judah.