I'd go through Rhode Island if you can do it. From there, you can go into Massachusetts and take the ferry to Orient Point in Long Island. The only down side to that is riding through Queens or Brooklyn. If you're okay with city riding, you'll be fine.

The best way to get to DC from there is to cross New Jersey and ride to Philadelphia. The East Coast Greenway offers a (mostly) navigable route through New Jersey, but it adds significant mileage to your trip.

However you get there, once you get to Princeton (or even Kingston), you can take the Delaware and Raritan towpath to Trenton, and then cross the Calhoun Street Bridge (there's a walkway) into Pennsylvania, and head south to Philly from there. However, this requires riding through Northeastern New Jersey, which is horrible for biking. I don't know the route offhand, but if you can cross New Jersey heading almost due West, you'll get to Frenchtown and you can the the Delaware and Raritan Rail Trail (different from the towpath) and cross into Pennsylvania wherever you like (there are a few bikeable crossings). This adds probably around 15-20 miles to your route, but it's a nicer ride. (You also lose out by missing Princeton this way -- a very nice college town.)