"His goose is cooked" means something has just happened that will cause something bad to happen to him. It is usually something he deserves. It means "he can't escape his fate now."
Ray: "His parents are gone for the weekend, so he borrowed his dad's car without his dad's permission."
Lewis: "He'll be in trouble if his dad finds out. But he'll probably get away with it. His dad will never know."
Ray: "Wait, I didn't tell you everything. He had a small accident and dented a fender. His father will notice."
Leiws: "Oh, boy. His goose is cooked now."