For example If I reside outside New York City, will I pay at both place?
No. If you reside outside New York City and you are exempt from the vehicle use tax in the county where you reside or have a business
I still need to find out about on how to register my vehicle in New York, what do I need?
You'll need a letter from the leasing company giving you permission to register the vehicle in NY, a power of attorney from the leasing company that authorizes you to register the vehicle, and the signature of the leasing company officer in box 3 on the Vehicle Registration/Title Application or on the Registration Authorization.