Not much of his past is reveal. Except that was in a forbidden relationship with Ren.
It is mentioned by Akito Sohma, that he was kind, and always smiling. Not much about his personality is given in the manga, or by Natsuki Takaya.
Fruits Basket Manga
When Akito was conceived, Ren threatened to abort her if Akira did not consent to raising her as a boy. Akira, afraid that Ren would keep her promise, agreed. Akira loved Akito as much as possible while Ren rarely looked at her. Akira once asked Ren why she refused to hold Akito, and Ren replied that she was more important than Akito. When Akira died, only Akito was present.
Ren came as soon as she could, but by that time Akira had passed away. Ren shouted that she should have been there, rather than this worthless child (Akito). Both Ren's claims of the bond being fake and Akira's death may have caused Akito to become as twisted, controlling and dependent as she is.
Akira was Ren's first and only love. When she was younger, she was a maid in the Sohma house; Akira's maid. She would always be his maid, no matter what. But as time moved on, Ren and Akira fell in love, and he married her. Akira loved Ren more than anything. When they had their first child, Akito, Akira seemed to love Akito even more than Ren. This made Ren very jealous of Akito, thereby hating Akito. When Akira died, Ren became depressed and hated Akito even more. She not only hated the fact that her one and only love was dead, but now the one to become head of the family was Akito, the daughter she never wanted or loved.
He was kind and caring father. He loved his child more,than his wife. When he was dying,he wife did not see him. But only his child did.
- He doesn't appear in the Anime.
all information on Sohma Akira came from http://fruitsbasket.wikia.com/wiki/Akira_Sohma