Did the Clones become the Stormtroopers?

Yes, the Clones did go into serves for the Empire, but because clones age faster then most humans, most of the Stormtroopers where regular men.

Yes, after Order 66 (the order from the chancellor that commanded the Clone Troopers to kill all Jedi because they presumably turned against the Republic), the Clone Troopers began serving under Darth Sidious, who renamed the Clones as "Storm Troopers." Eventually, the Clones were replaced Von normal humans that were recruited into the army.