Why did the ancient Jews believe the messiah would be a great warrior who would establish the kingdom through military conquest?

The ancient Jews believed the messiah would be a great warrior who would establish the kingdom through military conquest because the people's expectation of a savior became bound up with the notion of a new nation, political kingship.

What are a few ways that the Jews had already seen God's kingship?

A few ways that the Jews had already seen God's kingship was in the "wonders of creation" and in the Torah.