What are Paul's characteristics and motivations in "The Rocking-Horse Winner"?

Paul longs for his mother's love. We learn early in the story that his mother "felt the centre of her heart go hard" around her children. She tries to hide this, but her two daughters and Paul sense she doesn't really love them. 

Paul's mother, a proud woman, yearns for more money and likes to keep up appearances. She feels disappointed that her husband has not been more successful. Paul, a sensitive child, feels an "anxiety in the house" that haunts it. The very walls seem to cry out, "There must be more money." Everyone feels the "grinding sense of the shortage of money."

Paul internalizes his mother's desire for money. Like her, he is proud. He wants to please her, but his pride is injured when she doesn't believe that God told him that he was lucky: 

The boy saw she did not believe him; or rather, that she paid no attention to his assertion. This angered him somewhere, and made him want to compel her attention. 

Later, he again reveals his pride and desire to be taken seriously:

 And then the house whispers, like people laughing at you behind your back. It's awful, that is! I thought if I was lucky—

Paul is also secretive: "He went about with a sort of stealth, seeking inwardly for luck." When he wins 5,000 pounds betting, he doesn't want his mother to know he is the source of the money. He lies to her about why he has his rocking-horse moved to his bedroom.

He is an angry child too, as might be expected from someone who senses his mother really doesn't love him. We see this in his eyes. For instance, "his eyes had a strange glare in them." His eyes "glare" and "blaze." His voice shows his anger as well: he speaks "fiercely" and at one point his voice "flared." He rides his rocking-horse "furiously." We see his anger in his determination to "force" the horse to do his bidding:

He would slash the horse on the neck with the little whip he had asked Uncle Oscar for. He knew the horse could take him to where there was luck, if only he forced it.

There is also something odd about this little boy. His eyes "had an uncanny cold fire in them." The word uncanny is used twice to describe him.

We also learn in the story that Paul is "frightened."

Despite being a proud, angry, driven little boy, we feel sorry for Paul at the end, for he is most of all a frightened child who dies trying to earn his mother's love and approval.

How is Paul's determination portrayed in "The Rocking Horse Winner"?

The definition of determined is to be stubborn, persistent, and to not give up.  Think of that definition in terms of Paul's determination to "be lucky," and to win money at the races, all in a sad attempt to win his mother's love.  He is incredibly persistent; he thinks about it all of the time, and rides his horse for hours on end, even through the night sometimes.  That shows a fierce determination to attain his goal--money, on the surface, but, love from his mother in the end.  In fact, it is his sheer determination in the end that leads to his death.  He is so ferverishly focused on finding the winning name for the Derby that he rides into the late hours of the evening.  His mother gets home at "one o'clock in the morning" to find

"her son, in his green pyjamas, madly surging on the rocking-horse."

He rides himself into the ground, literally, and "collapses" on the floor in a fevered state, before triumphantly declaring that the winning horse is "Malabar!"  He had ridden for so long, with such intense and unhealthy focus, that he wore himself thin, giving himself a fever, and in the end, dying.  He was determined, no matter what, to figure out the winning horses.

I hope that those thoughts helped a bit; Paul's determination was a powerful force in his life, that was unfortunately focused on a losing battle:  the battle for money to bring happiness.  Good luck!

How does "The Rocking-Horse Winner" reflect and develop the effect of determination?

The most important idea D. H. Lawrence develops about the significance of determination in "The Rocking-Horse Winner" is that no amount of determination will lead to success if it is directed toward false or worthless goals.

In the story, Paul is depicted as a loving and sensitive son, as well as a boy of great determination. He is particularly determined to use his unique and mysterious gift to help his mother, Hester. All these are good things, but they lead to tragedy because Hester is obsessed with money, leading Paul to direct his determination exclusively to this end. The only time the quality of determination is explicitly mentioned in the text, it refers to Hester's determination to increase her income in a meeting with her lawyer. This is described from Paul's perspective:

He knew the lawyer's letter. As his mother read it, her face hardened and became more expressionless. Then a cold, determined look came on her mouth. She hid the letter under the pile of others, and said not a word about it.

As he dies, Paul is thinking about money and his determination to provide for his mother. He rambles deliriously in his final hours:

Over eighty thousand pounds! I call that lucky, don't you, mother? Over eighty thousand pounds! I knew, didn't I know I knew? Malabar came in all right. If I ride my horse till I'm sure, then I tell you, Bassett, you can go as high as you like. Did you go for all you were worth, Bassett?

The only reason Paul cares so much about money is because his mother does, and he loves her. It is his determination to prove his love and improve her life that kills him.

What is Paul trying to prove and to whom in "The Rocking-Horse Winner"?

Paul wants to prove to his mother that he is "lucky." Even though he is not sure what "luck" is, he declares that he has it. He does this in part to comfort his mother, and in part to supplant the place of his father, who his mother has called unlucky.

The story is told as a kind of fairy tale, almost. The home Paul grows up in is a kind of endlessly hungry monster which

+constantly demands more money. The mother is ashamed of their status as the "poor members of the family," dependent on relations for the use of a car. "Luck," the thing that the mother says "causes you to have money," becomes the key to a happy household, and Paul is determined to provide it, whatever it is.

Paul, as a child, conflates the imaginary and the physical. He rides his rocking horse to "where there is luck," a kind of place of psychological and emotional certainty or clearness that from which emerges the names of winning race horses. This certainty becomes tangible in the form of the thousands of pounds he wins at the races. Yet this tangible symbol of his luck is not enough to satisfy his mother. She circumvents his plan to give her a thousand pounds a year for five years by demanding all the money immediately, and this sudden influx of cash only makes the house demand even more money.

Paul is never able to satisfy his mother. She remains emotionally unreachable, as much a victim of her greed as Paul is a victim of her "luckless" state. Even when he wins an enormous sum, 80,000 pounds, essentially killing himself through the exertion, she remains unmoved.