I've made this trip a few times (driving all the way from central Virginia to JFK to take advantage of extra-low fares or mileage run deals) and I'd allow 5 hours, at a bare minimum, from anywhere in the DC area to JFK.

Assuming no unusual traffic delays, downtown DC to midtown Manhattan by car usually takes me 4 1/2 hours via the BW Pkwy/I-95/I-895/I-295/NJTP/Lincoln Tunnel route, traveling 5-9 mph over the posted speed limits and allowing one quick bathroom/food/fuel stop along the way. JFK is a good 20 miles or so east of Manhattan, so even cutting across Staten Island on I-278 it'll still take you longer to get to JFK from the NJ Turnpike than it will to Manhattan.

I'd recommend leaving DC by 4:30 a.m. just to play it safe and allow plenty of time to park the car at JFK and check in for your flight.