No, you won't use Clay credits when using your own Claude API key. When you connect your own API key for any data enrichment tool (including Anthropic/Claude), you can use that service free of Clay credits.
Instead, you'll be billed directly through Anthropic based on your subscription with them. This applies to other AI services too, like OpenAI.
To add your API key, go to your profile picture > Settings > Connections section.