How do I claim a pseudonym?

You can't!

Just become mega famous and nobody will want to use your nickname.

I might want to call myself Sting, but I'd just be seen as a dick and nobody would take me seriously (open goal for Sting jokes right there)
This would of applied to the days he was in The Police simply as a musicians nickname.

There's a wrestler called Sting, so it works because it's a different field of activity.

If you call your band or go as an artist as this nickname (for example Prince or as it happens now he's a solo artist trading as "Sting", Sting) then it becomes a protected trade name which is very different to just having it as a nickname.