You can obtain your certificate from the City Hall or the County of the city where you were born.
There are two forms of Birth Certificates. The Short form which has your basic information, and the Long form which is usually a copy of your original birth certificate as it was filed with the authorities. Unless you specify, you will be provided with the Short form. You will need to ask for the Long form to be valid in Greece.
The Birth Certificates issued by the Hospital are not useable. You will need to obtain the official birth certificate from the Secretary of your State.
The birth certificate must be issued within 12 months from the time you need to use it. Since you are applying for recent / current birth certificate, request a few extra copies (3 – 5). The cost for the additional copies is minimal.