We see Eddie as a tragic character because when we meet him, we want to like him and root for him. He is our protagonist, as evidenced by Miller giving him many lines and keeping him on stage constantly. Miller also establishes a Greek chorus in Alfieri and shows...
us Eddie's hamartia, or fatal flaw, in the first scene: his love for Catherine.
Eddie is a decent guy. He's "hustled" and worked hard for many years to provide for his family. He's taken in his niece Catherine and protected her:
"I promised your mother on her deathbed. I'm responsible for you. You're a baby, you don't understand these things."
We soon see that Eddie is not simply an overprotective relative; he seems to have developed deeper feelings for Catherine, even if he can't recognize it. Others do, though, including Alfieri.
Alfieri: "She wants to get married, Eddie. She can't marry you, can she?"
Eddie: (furiously) "What're you talking about, marry me! I don't know what the hell you're talking about!"
Eddie has "too much love for the niece," which is not right. But Eddie does not recognize this because he has suppressed these feelings. Eddie has created a bubble for himself, but by not confronting his feelings he continues to attempt to protect Catherine, which leads to his downfall.
Eddie may do bad things in the end, but he's not a bad guy. Miller crafts him to be a decent man whose motives are strong, but flawed.
We can also consider Miller's intentions by looking at his essay "Tragedy and the Common Man," in which he states
I believe that the common man is as apt a subject for tragedy in its highest sense as kings were.
It is time, I think, that we who are without kings took up this bright thread of our history and followed it to the only place it can possibly lead in our time—the heart and spirit of the average man.
Eddie is very much a common working class family man with a fatal flaw.