He visited The Early Show Monday and told co-anchor Harry Smith it's a story line that veers between drama and comedy.
"One of the nice things about cable is you can just write whatever story line you're telling," he said. "If it happens to be quite funny that week, that's fine. Or if it's dark that week or whatever combination, you know, wherever the story takes you, you can go."
One of the story lines involves Huff's brother Teddy, a schizophrenic who has just spent four years in a mental hospital.
"We really explore what his illness cost him. You see him really almost get it together. Then how tragic it is when he just can't," Azaria told Smith.
The show co-stars Oliver Platt, who has been Azaria's friend since their college days at Tufts University. Platt plays the part of Huff's lawyer and friend, Russell, who makes some questionable decisions in his personal life, like sleeping with Huff's mother at the end of the first season.
"He's amazing. He plays an alcoholic, drug addict, sex addict attorney," Azaria says of Platt.
The show received seven Emmy nominations last year. Its all-star cast includes Blythe Danner and Swoosie Kurtz and it has attracted some additional star power.
"Sharon Stone called and said, 'Can I be on the show?' We said, 'Sure, we'll figure it out.' And we wrote a part with Angelica Huston in mind," said Azaria. "And she said OK."
Azaria also stays busy doing a variety of voices, and estimates he's done 50 to 60 different voices over the years. Among them are several characters from the Fox series, "The Simpsons," and Azaria has started work on the upcoming Simpsons movie.