Find this book:
Relocating to New York City and Surrounding Areas by Ellen R. Shapiro.
She's very detailed. Even attempting to be as prepared as she outlines, my move was super easy. =) She covers expenses and what you need to get an apt, plus how to get a job. It's great.
My best friend and I prob saved about $10,000 combined--so $5000 is a good round-about amount. This should include first and last month's rent, utilities, broker's fee (if you need one), plane trip for scouting apts(we stayed with a friend), moving truck across country and all related costs plus some padding for the time before getting a paycheck. This is based on paying about $500 - $600 a month rent in Queens or Brooklyn. I highly recommend Astoria, Queens. =) It is going to be even cheaper if you don't have to do a moving truck--if you can just hop on a plane with luggage, perhaps mailing a few things.
I would def use craigslist.org to find an apt---it lists apts by whether they have a broker's fee and other search criteria--and it is free.
Good luck!