The clown first appeared in a British circus.
The first clown appeared in a British circus.
Which is correct if I'm talking about the history of this profession?
You need to rewrite your sentence, though the use of 'the' will remain appropriate.
Your sentence as it stands suggests that 'the very first clown in the world' suddenly came into existence with an appearance in a British circus.
More correctly, the 'first circus clown' (Billy Buttons) appeared in Astley's circus, which was a British circus.