They have announced the West Side Tennis Club schedule. The club will once again host the Sony Ericsson Tour Championship from August 21 - 25. It's a final tune-up to the US Open and gets famous players. There's also a few other tourneys of interest including a $50k Clay Court men's tournament in June that's open to everyone.
Most of these events are open to public viewing. I highly recommend going to a couple of matches this summer if you're around. No other area in NYC has anything like this and it's good to support it. Plus, the clubhouse and pro-shop will make you feel like you're on vacation in Cape Cod.
Here's a good NY1 piece about the stadium from 2004. They were talking about refurbishing it, but I don't think they've done anything with it yet. Tennis Fans Team Up To Restore Forest Hills Stadium To Former Glory
Joining the Club is expensive, but not as bad as a country club. You're looking at about $6 or 7k per year per person for tennis and pool, with monthly minimums on food at the club's restaurant. I've never joined, but I hope to someday. Where else in the city - or anywhere else for that matter - can you play grass court tennis on impeccable grounds?