I'm full Greek too, but 6 years at Greek school didn't really teach me much with the language. More about the culture etc...
However, really, the only respect in which knowing Modern Greek helps in learning Classical Greek is the alphabet. Otherwise, it's pretty intense anyway Alot of private schools offer it, but rarely are they taken. The schools that reliably have classes every year are Grammar and Pymble. Otherwise, usually there are kids from SBHS (my school, only public one offering Greek atm I think), and the following schools are on and off with classes: Trinity, Shore, All Saints Greek Grammar School, St Ignatius, Meriden, and that's all I can think of.