Hellerup is a great area close to the sea - only a few kilometers from central Copenhagen, and very easy to reach by train from the airport. Personally, I loved this combination of staying in a green, quiet place... and still, close to all happenings in Cph.
It´s a charming, old house, and a wonderful loft apartment!
NB, steep staircase - better to travel travel light :))
Nice coffee shops with great breakfast, almost next door.