Hitler had set orders that his body be cremated upon his suicide, his body was burnt in the Reich Chancellery outside the bunker. Soviet records show the remains of Hitler and his wife Eva were buried in unmarked graves outside of Berlin

In 1970 the remains were interred and burnt again afterwards scattering the ashes.

So no, there's no evidence to prove Stalin or anyone else used Hitler's skull as an ashtray.

Source: Kershaw, Ian (2008). Hitler: A Biography. New York: W. W. Norton & Company.

No. According to the head archivarius of the FSB there is a piece of his skull with the bullet hole in the archives inside of the Lubyanka building. This might be the source for the strange ashtray story. The rest of the Hitler cadaver was burnt.

source: times article behind a paywall and a Russian translation of the said article available for free.

I want to add that there is still discrepancy with whether he shot himself or not. All the German witnesses claimed they heard the shot and saw it after the fact. But the soviet doctor who performed the autopsy said that while a piece of the skull was missing that it is common when burning the corpse and the brain was completely intact which is evidence against the shooting. They found cyanide compounds in his stomach so they ruled the CoD as death from cyanide. The Russians thought dying by gunshot was too noble and wanted him to look like a coward so there is reason to believe the Russians tweaked the autopsy results.

Source is world at war "Hitlers two deaths" episode but I will try to find more academic sources later.