Byakuya was very hot-headed when he was younger, something that his grandfather Ginrei disapproved of. Since his days in the Academy and becoming a Captain, he has matured quite a lot.
He sees it as his role to set an example to his subordinates and thus rarely shows emotion and is very stoic a lot of the time.
We see his regret in how Rukia insists, early in the story, that he never looks at Rukia. Many of the clan have insisted that this is because she reminds him of Hisana, who we often see him lamenting over.
While we know Byakuya is still grieving his dead wife, we cannot say that this is the only reason that the captain is serious all the time. It is known that he also takes his position as the Head of the Clan and Captain very seriously as well.