According to an update on the official blog, Naxxramas will update with one wing at a time for five weeks. The Arachnid Quarter will come first, followed by Plague Quarter, Military Quarter, Construct Quarter, and Frostwyrm Lair. If you access Naxxramas during the launch event--roughly a month--you'll get Arachnid Quarter for free. It houses three bosses (Anub’Rekhan, Grand Widow Faerlina, and Maexxna), each of whom will add cards to your collection after beating them.
The other four wings can be purchased for 700 gold or $6.99 apiece. You can also purchase them in bundles set at different prices depending on how many wings you already own. If you want to purchase them all at once, it will be $24.99, but if you took advantage of the launch freebie and only need to buy the four remaining wings, it will be $19.99. Blizzard notes that purchasing the wings doesn't grant you early access, so you'll still be getting the wings as they release on a weekly schedule.
Naxxramas will also include a "Heroic Mode" of each wing, with more difficult bosses. Defeating all of the Naxxramas bosses on Heroic Mode will unlock a new card back, perfect for bragging rights.