The instructions on ordering NYC death certificates and the restrictions on who can receive the "long form" death certificates with cause-of-death information or only the short form are on this page at the NYC Department of Health: Death Certificates
If you have difficulty finding the exact record you seek, look for other records that can supply the same information, using a checklist or record finder such as:
- Sources of Genealogical Information (posted by The Gene Pool on Rootsweb)
- FamilySearch Wiki: United States Record Finder
- FamilySearch Wiki: New York Record Finder
As you consult general research guides like these, keep in mind that New York City has different record-keeping than the rest of New York State, so advice about finding records for New York State often won't apply to New York City.
- FamilySearch Wiki: New York, United States Search Strategies for Locating Births, Marriages and Deaths subcategory 1900-Present
- FamilySearch Wiki: New York City Genealogy
You could also consider hiring a professional genealogist who is experienced in the area. See this answer for links and guidelines.