Alice is what's known as a termagant wife, which means that she's an overbearing, bad-tempered woman. (Sorry, Alice.) The whole overbearing wife bit is often played up in literature and film for comedic effect, but unfortunately, Biff and Alice's lives are far from a comedy: they don't banter so much as wound each other:
"Listen," he said. "The trouble with you is that you don't have any real kindness. Not but one woman I've ever known had this real kindness I'm talking about."
"Well, I've known you to do things no man in this world would be proud of. I've known you to – " (1.2.16-7)
In a tragic turn of events, Alice dies shortly after this scene. And she doesn't just have any old death – Alice's passing is a gigantic, gruesome metaphor.
Within an hour they had taken Alice to the hospital and the doctor had removed from her a tumor almost the size of a newborn child. And then within another hour Alice was dead.
Biff sat by her bed at the hospital in stunned reflection. He had been present when she died. Her eyes had been drugged and misty from the ether and then they hardened like glass. (2.2.4)
It seems like Alice's tumor (and subsequent death) symbolize the state of her marriage with Biff and maybe the state of all relationships in the novel. They had wasted their time and forgotten about their dreams – and everything had festered inside this woman until she couldn't bear it any longer.
Alice's death is actually the key part of her role in the novel (which makes sense since it happens pretty early on) because it acts as a catalyst for Biff to reconsider himself as a person. After Alice dies, Biff even begins adopting some of her habits as he tentatively embarks on an exploration of his gender identity. How do you think Alice would have reacted to the new Biff if she'd been alive to see it?