Chronicles mode covered one side of the story, and whenever people want to add to the wiki it will contain story as well.
However, quick run down:
Raidou and Shiden are brothers of the Mugen Tenshin clan
Hayate is brother to Kasumi and half-brother to Ayane
Raidou is father to Ayane, Shiden father to Hayate and Kasumi, Ayame being the mother to all three
It is rumored (don't know if DOAD brought this up) that Hayate died in his fight against Raidou, causing Kasumi to pose revenge, but a clone of him was created by DOATEC called "Ein" or "One" who then fled.
Hitomi took care of "Ein" as he trained in Karate in Germany after fleeing from DOATEC, and later found he was the leader of the Mugen-tenshin clan.
Hayabusa comes from the Ninja Gaiden franchise and is married to Aileen - a US Agent from the same series
Hayabusa also runs a curio shop of which Zack is his best customer
Bass (Pronounced like the fish) is father to Tina, mother is deceased
Helena (pronounced "Elena") is daughter to Fame Douglas, the owner of DOATEC who hosted the first event and was assassinated by Bayman at the end of the first event. Fame had numerous mistresses, resulting in Kokoro and Helena being half-sisters. Bayman was hired by Victor Donovan, a scientist for DOATEC who wanted to take control. Christie was an assassin who was sent to kill Helena/Helena's mother.
