Take the Goethals Bridge to Staten Island, then the Verananno Bridge to Long Island (beautiful Brooklyn, that is.) Jump on the Belt Pkwy Eastbound and you're headed right for JFK.
HOWEVER, if I don't want to fight the hassle of the traffic (and it can be a killer at commuter time,) I'd pick up Public transportation from staten island to newark airport at Newark Airport (EWR) and ride a bus to JFK. It's a lot easier on the nerves. There is also rail transportation available transportation from staten island to newark airport Newark right to JFK.