To unlock Azure Trio, beat the game once. You will then get a message from him after the ending, which will let you know he's been successfully unlocked.
To unlock DG 0, Other Sel, and Shadow, beat Doppelganger twice.
If you want to unlock a secret epilogue, beat the entire Forest of Pain quest. This involves reaching the hundredth floor of the forest and defeating Cernunnos. The epilogue concludes the game's story.
When you beat the game the first time through, you'll unlock five new characters: Shino, Kite, Azure Orca, Azure Balmung, and Bourdeux.
To unlock Natsume, head to the BBS after eliminating all of the Chaotic PKs. If you then go to any area in the game, you will find her in a Battle Zone.
You can get special "marriage" endings for all twenty-two characters in the game. To do so, you must first complete a number of side quests in order to earn a Gift Card for the "Marriage Event" over at the Island of Kings (where Tournament Emperors are crowned). The character you give the card to must have a full Affection Gauge, and you should save before giving the card to any one character so that once you've seen a character's special ending, you can reset it and see another's. This is the only way to see all twenty-two special endings, one for each character.
To unlock Tabby, access the Forest of Pain's fiftieth floor.