The Stacks blockchain is a separate blockchain. We use Bitcoin for some functionality (for example, miners will need to interact with Bitcoin for burning tokens to participate in our mining algorithm) but over time this might change, see our roadmap for details: https://blockstack.org/roadmap/
One way to think about this is that we're bootstrapping our blockchain by re-using Bitcoin's hash power and may start relying less on it as more miners participate directly on the Stacks blockchain.