Claude is available for free (with usage limits) if you register for an account. Then you can pay for a $20/month Claude Pro account as an individual.
You can also create a paid API account as an individual and use it via their API, which is almost certainly cheaper than the $20/month Pro account for personal use.
I use the Pro account because I really like the Claude Artifacts and Claude Projects features.
As sibling comments have said, you can pay per month to use the web interface, pay for use of the API via self-signup or you can even use them via openrouter.
Is there a good way to use the Anthropic models as a private person? It's my understanding you can only pay for them as a business.