I like Chong Hap Market in Jackson Heights- near all the Indian restaurants
72-11 Broadway
Jackson Heights, NY 11372
They have fresh Kimchi trays there. I also like there Korean style chicken wings which they sell prepackaged and fresh to go.
Flushing also has a bunch of supermarkets on Northern Boulevard which I'm sure are better than the market on 32nd Street.
New Jersey just outside NYC also has a pretty massive supermarket with a nice Kimchi selection called Han ah Reum at 321 Broad Ave Ridgefield, NJ 07657