There really aren't any cheap parts of Manhattan anymore, except maybe Washington Heights or upper Harlem and even then not always. Go to Brooklyn or Queens -- much cheaper, much more space.
I live in the Williamsburg/Greenpoint area of Brooklyn, one stop into Brooklyn (I can walk out of my apartment and be in the East Village in less than 15 minutes and in midtown in less than a half an hour). The neighborhood has the whole young artist feel the Village had a decade or so ago, only cleaner and safer, with lots of restaurants, bars, performance spaces etc opening left and right. I live alone in my own two bedroom apartment, with two terraces (one with a view of the Manhattan skyline), a washer dryer, a roofdeck and other amenities, on a quiet tree-lined block two minutes away from a beautiful park and I pay less than friends in Manhattan living in tiny studios smaller than my living room. Mind you, Manhattanites have unfortunately discovered the neighborhood and prices are beginning to skyrocket (one bedrooms are starting to go for $2000+, and two bedrooms for over $3000 in some cases -- thankfully I've lived here for a few years).
If you want to live almost anywhere in Manhattan, you'll almost have to have a roommate or two and unless you're extremely lucky, in most cases expect your share of the rent to be around $1000 a month. You might check Chinatown, Hell's Kitchen (which I think the real estate people call Clinton now), the upper east side (oddly, though it's one of the ritzier areas of town, I've had several friends find deals in the east 80's and 90's), and MAYBE the lower east side (though I doubt it). You'd be better off in Williamburg, Boerum Hill, Carroll Gardens or Fort Greene in Brooklyn or in Long Island City or certain parts of Queens Boulevard in Queens.
There are deals out there (check the Voice) -- good luck.
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