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:
Wein originally intended the claws to be retractable and part of Wolverine's gloves, and both gloves and claws would be made of adamantium. Chris Claremont eventually revealed that they were an integrated part of Wolverine's anatomy in X-Men #98 (April 1976).
During the 1990s, the character was revealed to have bone claws.
Leading to the time when Wolverine had only bone claws, which actually lasted for over FIVE YEARS worth of Wolverine comics!!!
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:
The claws are made of bone, unlike the claws of normal mammals which are made of keratin, and were originally believed to be bionic adamantium implants.
Also, see the info on the Weapon X program:
Logan's bones, including his claws, were bonded with the indestructible metal, known as Adamantium, making them unbreakable, and was brain washed.
Logan's memories of much of his past would remain clouded, as a result of the Weapon X experiment and additional false memories during the procedure as well. Logan even believed that Weapon X surgically implanted his claws.
The last part being a sneaky way to Retcon them; since, up until then, canon was that they WERE implanted.
The bone claws are from his skeleton. They were always a part of his skeleton but he didn't know this since so much of him was wiped and/or ripped away. His skeleton had these claws attached to it but he just assumed they were implanted until Magneto ripped the adamantium out of him through every pore in his skin. Then realized they were actually bone, not metal, and certainly not the implanted weapons that they had all thought them to be.
Wolverine didn't have bone claws until the adamantium was ripped out of him. the idea of him always having bone claws is due to the writers ineptitude to know the characters actual history.
Case in point, the adamantium claws were due to the bonding process which had wolverine standing up in a tube with adamantium bands wrapped around him. the process used electricity to heat the adamantium bands to super hot temperatures which burned through his skin and onto his bones. Pretty much the reason why only he could survive while sabertooth couldn't was due to the rapid healing ability Wolvie had. The excess adamantium was forced out though his hands due to his healing factor which is also said how Wolvie can push them out using his healing factor.
The bone claws were the result of his healing factor going crazy to repair his bones that had the adamantium ripped off them. His healing factor, having had his adamantium claws for so long created the bone claws since his body was so used to having claws that it actually grew bone claws.
But then again some lazy SOB called Joe Quesada thought it would be best if they just said Wolvie always had bone claws because Joe knows how to make a great and compelling story.
So based on that said, the bone claws were the result of his healing factor going into overdrive to heal his badly wounded body and since his body adapted to have a system that allowed him to retract and pot out SOLID adamantium claws, the bone claws took the place if his lost SOLID adamantium claws. Which also caused his healing factor to burn out for a time due to the fact that he can't heal bone as quickly as soft tissue wounds.
Despite the movies being completely wrong about the adamantium process, wolverine did get the claws due to the process but it wasn't injected into him. He had bands of adamantium strapped to his body and they were super heated till they burned through his skin and laced his bones. The amount of bands he had on his arms was too much and so the healing factor pushed the remaining adamantium out through the back of his hands causing his adamantium claws to be created and also creating the method for him to actually operate them. the reason why they came out the back of his hands was because he was standing up when they did the process and due to gravity the healing factor pushed the adamatium out of wolverine through the path or least resistance.