Well here is the rule of thumb if you are going for lakes than go in July, Aug or September. If you are going for mountains than you can go even in June or end of May.
If you really want to enjoy the beauty of Kashmir than go after eid.
Otherwise Chitta Katha lake is awesome place and trek is not so difficult. If you are good trekker you can do it in one day. But I weill recommend you to camp at Duk2 and than on 2nd day camp at lake.