King David wrote a lot of them (something like 60 of them?) That would place them around 1000BC. As previously stated, some of them were written after Israel was placed in exile (you can tell because they talk of the destruction of the temple). This would place them sometime after 580BC.
The Psalter, which is the collection of Psalms as we know it today was assembled maybe around 200BC. And the Psalms as a whole were not considered inspired works until around 100BC.
I just purchased the Navarre Bible’s commentary on the Psalms this week, and I may have better information shortly.