Once Palpatine solidified his empire and began recruiting regular humans instead of clones, what happened to Kamino and its population? Did Palpatine order an attack on the planet? Were they left in peace (seems unlikely, when someone could approach them about cloning an army of insurrectionists)?
Shutdown following the formation of the Empire (19 BBY)
The only reference we get to Kamino's fate after the Clone Wars is in a side conversation a character has, which takes place shortly after the events of Revenge of the Sith.
"I heard they shut down the facilities on Kamino. They'll train up the last batch of clones and that's it."
The "clone rebellion" (12 BBY)
After the Clone Wars, the Kaminoans led a "clone rebellion" where they bred an army of clones to oppose the Empire.
"Officially, there never was a 'clone rebellion' on Kamino. Unofficially, approximately twenty years after we were created, a special detachment of the Imperial 501st Legion was dispatched to Kamino, with orders to eradicate an army of clones that had been bred to take arms against the Empire."
–Star Wars Battlefront II (2005), "Changing of the Guard"
As a result of this rebellion, the Empire greatly reduced the number of clone troopers used in their ranks and started using enlisted soldiers. This explains why Stormtroopers are less commonly clones.
Destruction from the Rebels (1 BBY)
The climax of The Force Unleashed II involves an assault on Kamino between Imperial forces and the Rebel Alliance. Since they don't have the means to hold Kamino, they choose to destroy the cloning facilities.
[Juno Eclipse asked] “What about Kamino? I hope you’re not thinking of leaving all this behind for the Emperor to start up again.”
[Bail Organa responded] “We’ll search the databases for any information on the space station the Emperor’s building. I’m sure it’s all been erased, but it’s still worth looking. Then we’ll ditch the Star Destroyers into the ocean and wreck the facilities with the resulting tsunami. In an hour or two, there’ll be nothing left.”
Starkiller’s expression was impassive, but she could tell that he was watching closely as the cloning towers receded below them. Barely had they lifted off when the first of the gutted Star Destroyers hit the ocean several klicks away. There was a flash like the rising of the sun as several megatons of water instantly vaporized. The shock wave radiated outward in a wave of steam hundreds of meters high. In seconds, the tsunami reached the towers, knocking them over and occluding the wreckage from sight.
–The Force Unleashed II (novel), "Epilogue: Kamino"
Financial collapse (after 4 ABY)
According to The Essential Atlas, the cloning facilities did continue is some capacity. I don't have a direct quote, so here's Wookieepedia's summary:
After the Battle of Endor, the collapse of the Empire triggered an economic collapse of Kamino and the production of clones eventually ended. The planet continued to survive on smaller contracts, often for warlords and criminal organizations.