New York is a city of neighborhoods, each with a slightly different character. Most if not all of it is photogenic in some way, and coming to it with your eye may be an asset--you may see things New Yorkers don't. I've always liked the lower east side and the Village for the architecture and the eclectic mix of people. As Peter said, some of the communities along the East River are interesting and the views of Manhattan from there are iconic. I like Dumbo, not least for the Powerhouse bookshop and gallery (they published the Vivian Maier book) but also for the shots you can get there. Brooklyn has many neighborhoods, as does Queens, and the two of them are vast. If you've only got five days, rambling around Manhattan and maybe a day in Brooklyn and Queens would give you a good taste. I have to say too that a trip across (under) the Hudson on the PATH train to Hoboken is good for scenic views of the city.
I used to live on Roosevelt Island; the aerial tram from 59th Street is fun and good for shooting.
If the weather is good, Bryant Park is interesting for people-shots as, of course, is Central Park. If you're going to B&H, keep walking uptown through Hell's Kitchen to the theatre district or all the way up to Lincoln Center.
You should visit Grand Central too--if you ask the security guards there you can get a permit to shoot which I think is free and is good for an hour or two.
And you probably should walk the High Line, even if it's full of strolling tourists.
You will, no doubt, find places that you like best just having a wander down this street or that.
