Ach Pear, it's a long and complicated story with bilingual schools in Berlin - having had kids in them for the past 13 years. Why don't you start out by checking the relevant page of Toytown for people's opinions, because for every person with a negative experience there's someone who loved that same school.
My immediate thought is that if you are British you haven't got a chance in hell of getting into the JFKS - in fact if you apply right now as a non-American or German, they don't even bother acknowledging your application. And it's not THAT great a school, it just has better resources than other non-private schools in Berlin, which are mostly horribly underfunded. There's more about the JFK that I'm certainly not going to say on a public website.
Are you talking about primary or secondary? Is your DH by any chance a journalist or in the diplomatic service? If so you have a better chance of getting in to the Nelson Mandela, which has places reserved for highly mobile families - but simply working for an international company doesn't count most of the time. Or maybe you could strike lucky, who knows.
Your best bet at primary level is to try one of the two Europa-Schulen - the Charles Dickens or Quentin Blake. At secondary level your choices are hugely restricted in state school if your DC have no German - you'll essentially be pushed into the Peter Ustinov (which is a comp / secondary modern) and refused access to the grammar school the Schiller, however clever your DC is.
However, there are quite a few private bilingual schools now set up across the city - and the fees are MUCH cheaper than on average for the UK. Berlin Cosmopolitan, Phorms, Kant-Schule, Anna-Sophie are all worth trying and have both primary and secondary. The atmosphere there tends to be much nicer.