During The Force Awakens we see Kylo Ren "talking" (almost "praying") to the half-melted helmet of Darth Vader.
How did he come by this item? Was it not destroyed with the rest of the Death Star at the end of Episode VI: Return of the Jedi?
No, it was not destroyed. It didn't burn in the fire (predictably, given what its design purpose was).
We don't know for sure from current canon.
But in the "Aftermath" prequel novel, we see that people collect Darth Vader related items (presumably, largely fake?), so this could have been a likely avenue.
“Is it really his lightsaber?” the young woman asks. Her voice is a raspy whisper, as if something is wrong in the well of her throat.
“It’s the Sith Lord’s laser sword, sure enough.”
“May I?” she asks.
The Kubaz shakes his snout and says: “No. Not until I see the money. Money talks or Ooblamon walks.”
Ooblamon’s little friend, the rat-faced girl, giggles when she translates. The pale woman looks to the other two in their dark robes. They whisper to each other. Almost as if arguing.
She turns back. “How do we know it is Vader’s blade?”
“You don’t. But it’s a lightsaber, isn’t it? And it’s red. Isn’t that the color you seek?”
The young woman offers her pale hand. “The…lightsaber, please.”
“When the count is complete,” Ooblamon says. He cocks his head and stares at them through his goggles. “What are you? You’re no Jedi.”
“We are adherents,” she hisses. “Acolytes of the Beyond.”
“Fanatics of the dark side?” he asks. “Or just children who want to play with toys?”
“Judge us not, thief.”
(Aftermath: Star Wars: Journey to The Force Awakens; "Interlude: Taris")
At the end of Return of the Jedi we can see that Luke saved his father's body including the helmet, and burned them in a funeral pyre.