You did your research and found your perfect neighborhood in New York City. You found a broker who understood your needs and then went out there and pounded the pavement. Finally, you found the ideal apartment. Now, it all comes down to this: submitting your lease application and getting it approved. Having the lease application documents ready ahead of time will help ensure you get into that dream apartment before someone else can.
The most important check that a landlord or management company will perform is on your ability to pay your rent. A few buildings will also check to make sure you're a good tenant (e.g., you won't solicit complaints from your neighbors), but unless you can send in your rent check on-time every month, you are going to be declined outright.
The first thing your new building will want to know is what percentage of your total income will be going towards your rent check. The general rule of thumb is that your annual income (pre-tax) should be at least 40x the monthly rent. In other words, if you want an $1,800 a month apartment, you will need an annual income of at least $72,000. It would be nice if your new building would just trust you on your income, but you'll need to supply pay stubs, a letter of employment, last year's tax return, driver's license, and potentially even your bank statement.
The next step for your landlord is to make sure that you can actually devote that portion of your income to rent. The building will run a credit check on you, which will tell them if you have any significant debts or if you have a history of not paying your bills. Before starting the process, make sure you take a look at your credit score (AnnualCreditReport.com for a free overview or buy a FICO report to get an exact score) so that you know what you're up against. If your credit is an issue, you have some options, but start by doing what you can to improve it.
If your credit score is poor or you can't cover the monthly income requirements, you aren't totally out of luck, but you'll need some outside help. Most landlords allow tenants to sign a lease as long as they are backed by a guarantor (someone who will be legally obligated to pay the bill if you're not). The restrictions for guarantors are higher than those placed on the tenant, both in terms of income (70x rent) and credit, but also in terms of location. In order to improve their likelihood of actually collecting the payment, the guarantor may have to live in New York or nearby in New Jersey or Connecticut.
One final note: just about every apartment will require a one month security deposit (that you get back at the end of your lease if you don't destroy the apartment) as well as your first and maybe even last month's rent. Be sure that you have the cash available to make those payments, which on top of a broker's fee, can be quite a large cash outlay before you move in.
Make sure you have all of these papers and requirements before you start looking, and even consider carrying them with you as you visit apartments. Although there are a lot of steps in the process, you can go from applying for the lease to keys in hand in less than an hour if you're well prepared. Having all of your ducks in a row can make all the difference when it comes to renting the perfect New York City apartment.