I need to file a small claim in Buffalo, NY and need to know the procedure.

Not sure on NY law but here in WI, you take copies of all your supporting evidence(reasons why you're suing) to the courthouse and fill out the paperwork. They need the copies to provide to the defendant when they are served.

You'll also need to have the defendant's physical address so that they can be properly served the paperwork by the police.

Sort of the same - you go to the jurisidiction of the Defendant (which may or may not be the City of Buffalo or a suburb); you fill out one page listing your name, their name, the cause of action, the dollar amount. No need to provide any "proof." Pay the Court fee. Usually you get a notice in the mail when it's scheduled; some Courts schedule the case when you file it.

Pretty much the same all over NY.

THIS SHOULD HELP:

A Guide to Small Claims Court