Check Eldridge for the slack times, you'll find it at every boating store. If you are at a yacht club on CI, odds are someone else at the club has one you can take a look at. Make sure your engine is reliable, sailing in the East River is allowed but not suggested unless you really like short tacks. You've got about a two hour slack (slack-ish) at the Hell Gate, and outside of that slot you can hit currents over 6 knots. I spent an hour with some friends "racing" under full sail and engine and making a whole hundred yards north of Astoria that way one day, someone got stuck and we had no choice but to either try it, or wait.
Usually there ARE special regs for the 4th of July and your best source of information is the USCG directly. Any local group can either tell you what the regs or, or who to contact. Typically, the East River and part of the Hudson will be closed to all traffic from sunset on the 4th until midnight. There are also exclusion zones (i.e. you can't pass alongside the UN building at any time, you must keep clear of naval vessels) special designated anchorages, etc.
If you prepare for the timing, a possible need to wait/anchor out, etc. it can be a great way to watch the fireworks. You'll also find every yahoo that can get on the water may come out, so you need to keep watch on what's happening around you. "Sheltered" locations inside the bulkhead
line are usually private, and there's an open anchorage between Ellis and Liberty (IIRC) but that will fill early and you will rock there.
With food, drinks, sunblock, and lots of patience...a great place to be.