The one year we we went the second weekend in April it was cold. Chilly enough to need a hoodie at all times. It was very windy and too cold to spend any considerable time on the beach. We had a good time, but said we'd wait until the end of April/May to go again.
It's chilly! If you're in NJ as your username implies (we are, too!) it's about 10 degrees warmer than it usually is here in April.
The ocean water is still way too cold to go swimming. If you're trying to go during the off-season, after Labor Day is better as the ocean water is still warm (relatively... ).
Oh boy, I can answer this one since I live here! The ocean water will be freezing cold. However, the air temperatures will probably be pretty nice provided the wind lays out. April is a good time to visit for fishing, shelling and relaxing. Look for a hotel with an indoor pool and you'll be all set!
We went in early April once. It was windy and chilly. The kids' kites flew away, and we had sand embedded in our scalp on the beach (where we flew kites).