You provide no information as to why you are going to the appellate court. Why are you not going through family court for custody? If you are appealing a family court decision - you would have to file a notice of appeal within the prescribed time period - then following the appellate procedure for your department. Go to the appellate division website and download the local rules which will detail what must be included with any appeal. There are specific rules regarding the number of copies that must be filed, specifics for the appendix, the format, etc.
You should consult a lawyer about your appeal. If you can't afford a lawyer you should read the New York State Bar Association's Pro Se Appeals manual at the link below.