Were Wolverine's claws formed as part of forming his Adamantium coated skeleton?
Or were they part of his mutation?
No, it was established in Wolverine volume 2 #75 that he has had bone claws all along as part of his mutation. They were always thought to be part of the adamantium coating process. This was disproven after Magneto ripped all the adamantium out of his body (X-Men volume 2 #25, 1994), leaving him with bone claws.
His origin mini-series, titled Origin, depicted him with bone claws as well.
When Magneto stripped the adamantium out of Wolverine it was revealed that he had bone claws underneath (as depicted in X-Men Origins: Wolverine). However that was not always the case:
So while he was not originally intended to have bone claws, that is how his storyline eventually evolved.
The claws are part of his base mutation, as was seen when the Adamantium was ripped from his skeleton by Magneto.
From the Marvel page on him:
Also, see the info on the Weapon X program:
The last part being a sneaky way to Retcon them; since, up until then, canon was that they WERE implanted.