Unfortunately there is no direct flight from Dover to Calais. The nearest airport to Dover on UK soil is Kent International from which there are no scheduled flights to northern France. Similarly, the nearest airport to Calais on French soil is Calais – Dunkerque Airport which is used for small aircraft only.
If you want to travel by air across the channel to visit Calais, the best and cheapest way is to take a plane from London to Paris and travel by rail from Paris to Dover.
The main airports in London and the South East that service regular flights to Paris are Heathrow Airport in west London and Gatwick Airport in Sussex, south of London. From both airports, flights to Paris arrive at Charles De Gaulle Airport which is 17 miles North East from the centre of Paris in Roissy-En-France. The flight takes about 2 hours.
On arrival at Charles De Gaulle you need to take a train on the TGV Interconnexion line to Lille Europe which takes about 1 hour. At Lille Europe you have to change trains and have two options.
The first option is to take a TGV Interconnexion train from Lille Europe to Calais Frethun which takes 30 minutes. The second option is to change stations to Lille Flandres and take a TER train to Calais Ville which takes 1 hour 30 minutes. It is worth bearing in the mind the second option in case of unexpected delays or cancellations. The total journey time is at least 4 hours.