He is Julie’s father and the man in charge of the stadium’s pseudo-military government, General Grigio is a man living his own kind of undeath. In his youth, he was a dreamer, a musician, a wine enthusiast, but the collapse of civilization and loss of his wife have sucked out every drop of hope. He retreats into alcohol and brutal pragmatism, spends his day's building walls and training soldiers, his nights drowning any surviving emotions in whiskey, despite having declared alcohol illegal.
He devotes all his energy to keep his people alive but sees no reason why life is worth living. He loves his daughter, but she may not be enough.
He holds a very firm opinion on the idea all zombies are lost causes and need to be shot. Zombies are the reason his wife is dead. She got bitten, which soon turned her into a zombie. He had to shoot his wife in the head. His daughter and his people are the only things he has left, and he is dead-set on protecting them.
As noted by Nora, Grigio has a sadistic love of killing zombies. This is so strong, he attempts to kill R, who proved zombies were changing back into normal humans, just because he was one.