A witty British pun. :) Berkshire is a county in England's south east, but it's pronounced "bark-shur". So if someone is coughing a lot (sounds like barking), then they should run for local government in "Bark"-shire.
I think the phrase is still alive in the UK, but outside of there perhaps we (non-Brits) don't have such a close equivalent.
We sometimes say "coughing up a lung".