I live in NY...Queens to be exact. I went through this 2 years ago. My advice is to get up to at least 620 and clear as many collections as you can. Most of the buildings that use management companies tend to be a problem because they let a computer program deny/approve. Very rarely to they do full manual reviews unless they see you were borderline. Try to stick with buildings that you can deal with the landlord directly. I lucked up and ended up in a very nice neighborhood and I was honest about my credit situation. The landlord asked for one extra month security deposit and agreed to a one year lease as a trial basis. I paid my rent before the 1st every month and after the year they returned that extra security and now i'm in a 2 year lease. I had a TON of collections and about 5 judgments on my reports. I never had any evictions showing though.
As of 11/02/15:
EX-679, TU-707, EQ-690 (FICO 08),7/31/15 691 (FICO 04)
CAP ONE QS-$4,250, CAP ONE QS-$2K, CAP ONE VISA SIG-$7K, JETBLUE AMEX-$5K, DCU PLATINUM VISA-$6.5K, AMAZON STORE-$5k, BANANA REPUBLIC-$5k
AU ACCTS: CAP ONE QS-$1.5K, CAP ONE CLASSIC-3K, BLOOMINGDALES-$3K, NORDSTROMS-$1K