How is William Faulkner's "A Rose for Emily" a story about the human heart in conflict with itself?

There is certainly plenty of conflict in this excellent
short story, both internal and external. However, the conflict that I think your
question alludes to is the internal conflict that goes on in Miss Emily herself, a
character that we only get to see through the eyes of other people and never from her
own perspective. We are but given tantalising glimpses of her background, that mostly
focuses on gossip and hearsay, however, one distinct memory that we are given is the
strict and authoritarian way in which her father raised her, scaring away any suitors
with his whip:

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We had long thought of them as a tableau, Miss
Emily a slender figure in white in the background, her father a spraddled silhouette in
teh forground, his back to her and clutching a horsewhip, the two of them framed by the
back-flung front door.

It is
clear from this "tableau" that Miss Emily had a very difficult childhood in a sense, and
was kept isolated from forming human attachments with others thanks to her father.
However, as the story progresses, we continue to see how Miss Emily becomes ever more
stranded from the rest of humanity, and how, in a sense, she is already living a life
that is dead, as her description as a "bloated" corpse, "like a body long submerged in
motionless water," indicates. As we discover the grisly murder that she committed we
realise the conflict that Miss Emily has undergone. Having had love denied to her for so
long, she finally receives it, only to face losing it. Her desire for companionship of
any kind, whatever the price, is evident in the strands of grey hair that adorn the
pillow next to the corpse of Homer Barron. Through killing him, she gains him, but also
marks her own exit from the world through her corpse-like appearance. Love is so strong
sometimes that it expresses itself in acts of violence, and even of
murder.