Train is the cheapest and maybe the quickest, (need to change in Benidorm) then a taxi from Alicante city centre to the airport.
Other option is a taxi all the way, which thinking about it will probably be quicker as no stops/changes http://www.idealspain.com/pages/information/train_alicante.html
I'd say Valencia is closer than Alicante airport to Denia, if not then there is not much in it.
There is an Alsa bus to the airport definitely from Javea, so probably also Denia. Or you can get a taxi to Xeraco Station about 5 miles up the N332 and then its about €6 to Valencia on the train and then a short metro ride direct to the airport.
As others have said a one way car hire may be easiest if not cheapest