Here's the breakdown:
* Wolverine's claws are actually bone. They are an extension of his skeletal structure, much like the claws of a real-life animal.
* The adamantium is a coating. It's a near-indestructible metal that was bonded to Wolverine's bones, including his claws, during a Weapon X experiment.
So, while the claws are part of his skeleton, the adamantium is a separate material that covers them.