How does Conor's relationship with the monster change throughout the course of the narrative in Patrick Ness's A Monster Calls?

At first, Conor believes the monster is merely a vision in a dream; then, he becomes confused about what is real and what is not. Further, he does not understand the meaning of the monster's three tales until he admits the truth about himself, a truth which the monster helps him realize.
When the monster first calls, it appears after the moon passes before Conor's window that looks out upon an old church and its graveyard. Also visible is a large yew tree that transforms into a monstrous human shape:

I have come to get you, Conor O’Malley, the monster said, pushing against the house, shaking the pictures off Conor’s wall, sending books and electronic gadgets and an old stuffed toy rhino tumbling to the floor (Chapter 1).

Conor refuses to believe this monster is nothing more than a dream. After Conor is attacked by the bullies at school, the monster reappears that night at precisely 12:07 a.m. When the monster tells Conor that he wishes to talk with him, Conor says that he will meet him outside so his mother will not be awakened. Conor is amazed that he walks down the stairs when it is only a dream.

“It’s only a dream,” he said again.But what is a dream, Conor O’Malley? the monster said. . . Who is to say that it is not everything else that is the dream? (Chapter 5)

The monster tells Conor that he only appears for matters of life and death; further, he states that he will return to Conor, and he will relate three stories from when he walked before, and they will be scary. Also, after he tells three stories, Conor will tell a story, and it will be the truth--not just any truth, but Conor's truth, the truth of which Conor is afraid. 
Later, the monster returns and relates the first of his tales, a story about a man who thought only about himself. The tale involves a kingdom that experiences hardships, as all the males die but the grandson. Conor does not like this story because the surviving grandson kills his beloved farmer's daughter and blames it on his grandmother, the regent, so he can be king when he becomes old enough.

Conor hesitated, confused. “You said you made sure she was never seen again.”"And so I did. When the villagers lit the flames on the stake to burn her alive, I reached in and saved her" (Chapter 9).

The second tale is about a man punished for his selfishness. He was a minister who preached against cutting down a yew tree for medicinal products, and drove an apothecary away. When his daughters were ill, the minister begged the apothecary cure his daughters. The apothecary refused; consequently, the girls died.
Conor argues that the apothecary should have helped. The monster responds,

He was greedy and rude and bitter, but he was still a healer. The parson, though, what was he? He was nothing. Belief is half of all healing (Chapter 17).

The third tale is about an invisible man. He is not truly invisible, but people no longer pay attention to him until the man finally decides to make people see him.
All of these tales are relevant to Conor's life. The yew tree which transforms into the "monster" gives advice and teaches lessons with his three tales. He wants Conor to come to grips with his mother's terminal condition and stop trying to be "invisible" because, he tells Connor, he must tell the truth:

Belief is half of healing. . . I came to heal you (Chapter 24). 

Conor has thought more of himself and his grief over his mother's approaching death. Rather than meaning to frighten Conor as in their first encounter, the monster becomes an instructor to the boy:

You must tell the truth or you will never leave this nightmare, the monster said, looming dangerously over him now, its voice scarier than Conor had ever heard it. You will be trapped here alone for the rest of your life (Chapter 27).

Finally, Conor admits the truth: that he has wanted to believe that his mother would be cured. The monster tells him that he has not come to heal her, but rather to heal him by encouraging Conor to tell the truth.

And it was for this that the monster came. It must have been. Conor had needed it and his need had somehow called it. And it had come walking. Just for this moment (Chapter 29).

In the final chapter, Conor tells his mother that he does not want her to go. He holds her in understanding until he can let her go.