International travelers fly into Noi Bai International Airport in Hanoi or into Tan Son Nhat International Airport in Ho Chi Minh City to get to Hue.
Air France is one of the major carriers servicing Hanoi and Ho Chi Minh City . Tiger Airlines service Hanoi and Singapore. Other airlines with regular service to Hanoi are Malaysia Airlines and Korean Airlines (every Asian airlines service the country). Vietnam Airlines offers some interesting deals to travelers as cheap ground transportation to locations in Hanoi. It operates flights from Berlin, Hong Kong, Kuala Lumpur, Manila, Los Angeles, Moscow, Paris, Seoul, Singapore and Sydney among other destinations in Europe and Asia.
Noi Bai has been recently remodeled and many travelers prefer it to arriving in Tan Son Nhat (but in some cases you can get lower fares if arriving in Tan Son Nhat). The departure tax in both Hanoi and Ho Chi Minh City are $14. They accept USD and VND.
From Hanoi, you can take either train or bus to get to Hue. If traveling by train, take the Express Train. It is faster and more comfortable, more expensive though (slow train around $15, the Express $55). Ho Chi Minh City is connected to Hue by road and rail as well.
Hue has an airport ( Phu Bai Airport ) that receives domestic flights. Perhaps it is your best bet as flights from Hanoi are cheap and in 45 minutes you'll be in Hue (against the 15 hour trip on the slow train).