Alfieri functions as the chorus of the play, commenting on the action and the characters carrying out the action, but ultimately set apart and serving as a medium between the audience and the action of the play. 

...it is Alfieri’s view that defines the action of the play and its unfolding.

Alfieri opens and closes the play with a poetic monologue that functions as a frame story. Within the context of the play, he narrates certain sections to show that time has moved on and also serves as a moral sounding board (almost like an oracle) when Eddie finds himself as a moral crossroads. 

Though Alfieri is unable to change the course of Eddie's behavior, it is his role to explicate this behavior, communicating the meaning of the play's tragedy to the audience. 

Alfieri serves as spokesperson for all as he delivers the final monologue, bringing the tragic tale to a close.

Without Alfieri, the play's thematic and formal connections to the plays of ancient Greece would not be clearly articulated. Alfieri then functions as a chorus, a narrator, and a formal link between Miller's inspiration and his own work.  

Discuss the character of Alfieri and his importance in A View from the Bridge.

Alfieri is one of the strangest characters in all of Miller's works. He acts like a Greek chorus in the play, and his viewpoint is the "view from the bridge" between the US legal system and older, cultural or tribal 'law'. He is an Italian-American, a student of American law, but still loyal to old Italian customs.  In more than one way, he's a combination of the old and the new.

Therefore his view of the interaction between American Carbone and his Italian relatives purports to be objective, and even-handed. The big question about him is whether it is or not: which provokes the same question about the potential bias of the play as a whole.

Alfieri breaks the fourth wall and talks directly to the audience. It seems he tells the story which the play re-presents: and he comments on the events and the characters regularly. He's actually quite inconsequential to the action of the play as a character, but hugely important as an alienating reminder to critically judge the characters and events we are watching. By ruining the illusion of reality, we, like Alfieri, powerlessly watch the events unfold, and try to understand what this modern fairy tale might show us.