GameCube controllers cannot be used to play The Godfather: Black Hand Edition on Wii. The Wii version was designed exclusively around motion controls and does not support the GameCube controller, despite the console's backward-compatibility ports.
The Wii release's core design centered on motion-based combat mechanics branded as the "Black Hand" system. Players physically mimic strangling, shoving, and aiming with the Wii Remote and Nunchuk. A GameCube controller lacks motion sensing, an IR pointer, and the gesture recognition required for these mechanics. Retrofitting traditional dual-stick controls would have required a parallel control scheme the developers chose not to build.
Bottom line: Play the Wii version with Wii Remote + Nunchuk, or choose the PS2, Xbox, Xbox 360, or PC versions for traditional gamepad control.