Throughout The Hunger Games, Katniss goes through a number of changes, both physically and mentally. This is to be completely expected in light of her experience; she is thrust into a death arena where the chances of survival are slim to none. At the beginning of the story, Katniss is very hard-edged, a defense mechanism she has honed after years of living in District 12 and seeing the effect of this rough environment on her friends and family. She is afraid to forge relationships and entanglements with another person for fear that she will eventually lose that person. She is only really close to Prim and Gale.

But starting with this first novel in the trilogy, she matures and develops empathy for Peeta, Rue, and even Haymitch and Effie. This development goes hand in hand with another way she grows: learning to work as part of a team. She is used to doing tasks on her own or with minimal assistance, such as when she is hunting or taking care of Prim. Being in the arena forces her to work with Peeta and Haymitch in order to ultimately win the Hunger Games. As a District 12 "team member," she takes direction and shows her leadership qualities.

Physically, she becomes even stronger after the practice/training session before the tributes are thrust into the area and then during her time competing. She's running, hunting, hiding, climbing trees, and so on—all survival skills that build up her strength and endurance.

In Suzanne Collins' novel The Hunger Games, the protagonist, Katniss Everdeen, undergoes a number of changes over the course of the story. 
 
At the beginning of the novel, Katniss is fearful and unsure of herself. While she illustrates bravery when she volunteers as tribute in place of her younger sister, Prim, she doesn't think she's going to survive and has little confidence in her abilities. She feels almost like she's a pawn in the Capitol's games and she's basically already accepted that she's most likely going to die. 
 
Before she takes the train to the Capitol, she pleads with her mother to take care of Prim. Katniss reminds her that she'll be the only one Prim has once she dies. On page 37, Katniss even says, "I can't win. Prim must know that in her heart. The competition will be far beyond my abilities" (Collins, 37).
 
However, once she's actually participating in the Games, Katniss is able to hold her own for a long time. She escapes the Careers by cutting down a tracker jacker nest but she's stung in the process and ends up blacking out. When she wakes up, she finds that Rue, a young tribute who reminds her of Prim, has tended to her wounds. Katniss takes a liking to her and the two form an alliance.
 
Things go well until Katniss and Rue decide to separate in order carry out a plan to blow up some of the Careers' food and supplies. After blowing up the food, Katniss is on her way back to their meeting spot when she hears Rue screaming her name. She finds Rue trapped in a net and before she can free her, a boy from District 1 plunges a spear into her. 
 
Katniss kills the boy and stays with Rue as she dies. Rue tells her that she has to win and Katniss responds, "'I'm going to. Going to win for both of us now,' I promise" (Collins, 230). At that moment, Katniss regains her will to fight, for both Rue and herself. She sings to Rue until she finally passes away and then covers her body in beautiful flowers, signifying a proper burial. Then, she salutes Rue with a "goodbye" gesture that's unique to District 12. 
 
By that point in the novel, Katniss has changed significantly. While she's still afraid, she's now angry. She no longer feels like she can afford to lose the Games and will do anything to come out on top for Rue. Though she's still technically a player in the Capitol's games, she rebels against them and everything they stand for by honoring another fallen tribute. 
 
She even says, "I want to do something, right here, right now, to shame them, to make them accountable, to show the Capitol that whatever they do or force us to do there is a part of every tribute they can't own. That Rue was more than a piece in their Games. And so am I" (Collins, 233-234). 
 
That said, Katniss is no longer the cynical, passive girl she was when the novel first started. Now, she's angry, determined, resilient, and most importantly, defiant.

In The Hunger Games how does Katniss develop from the beginning to the end of the book?

Considering The Hunger Games, the first book of the trilogy, alone, it is in fact difficult to see Katniss's development. However, she is not the same person at the end of the Mockingjay, and in retrospect, the reader can see that the changes began during the events described in the first book.

As mentioned in the previous answers, Katniss is not afraid to break rules and she is single-mindedly focused on her and her family's survival. Even her volunteering to take Prim's place in the Games might have been more due to practicality, rather than sentimentality—Katniss is older and more likely to survive. This is not surprising given that survival has always been a struggle, and there was simply no time, or strength, for anything else.

This starts to change during the Games, when the struggle is not simply to survive, but to survive without becoming a murderer. At first, when Peeta mentions to Katniss the night before the Games that he does not want the Games to change him—to make him a savage—she is annoyed. But as she meets other tributes, particularly Rue, she begins to understand and relate to what he meant. So, her world is no longer black and white. She develops connections to people outside of her family. She realizes that Peeta has known her for years—she has been "a subject of dinner conversations" in his family—and her actions had left imprint on the residents of District 12, something she never considered before. 

As the series progress, Katniss becomes less and less judgmental and narrow-minded. She becomes much more forgiving of other people's faults, such as Haymitch's drinking and the silliness of her "prep team." She begins to try to forgive her mother, whom she resented since she was too depressed to care for her children after the death of Katniss's father. While in District 13, she even develops friendship with Johanna Mason, a tribute from District 7, who she originally despised. In other words, as the story evolves, Katniss becomes compassionate and perceptive. This perception might have been what enabled her to see Alma Coin's impeding takeover of Panem for what it was, and change the course of history single-handedly by killing her.

I must agree with deem1510 in that Katniss does not sprint across the finish line but limps through it. At the end of the trilogy, having lived through the violence and the constant danger of the Games and the war, she is suffering from post-traumatic stress. But she still finds the courage to face each day, to help rebuild the District 12, and to raise children, despite her great reluctance to take this risk.

In The Hunger Games how does Katniss develop from the beginning to the end of the book?

If I am being honest, I do not believe that Katniss does change and develop over the course of the novel. She hates the Capitol and its power from the beginning of the novel to the end. She is rebellious throughout the entire novel. She refuses to play by the rules. We see that at the beginning when she is out illegally hunting, and we see it at the end when she forces the Gamemakers to allow two winners. She is brave throughout the entire novel, and she is more than capable of holding her own. She constantly shows an ability to adapt and think on her feet, and for the most part, she cares more for her own survival than anybody else's survival. It is something that Gale points out in the third book:

"Oh, that I do know." I can just catch Gale's last words through the layer of fur. "Katniss will pick whoever she thinks she can't survive without."

While Katniss is focused on her own survival, there are a few exceptions when she does consider the welfare of someone other than herself. The best indication is the Reaping. Prim's name gets chosen, and Katniss decides to offer herself as tribute instead. During the actual Hunger Games, Katniss is offering some protection to Peeta and Rue, but readers are often left suspecting her motivations. Is she really protecting them or just using them as tools for her own survival and manipulation of sponsors?

If I had to pick some kind of specific way that Katniss shows some kind of actual development, I would pick the part of the novel right before she and Peeta are going to commit suicide. It is at this moment that Katniss realizes that she can actually beat the Gamemakers at their own game. She realizes that she, as an individual, can fight and win against the Capitol's oppression. Up until this moment, Katniss does not believe any of her actions will make a difference in the overall state of the world:

We both know they have to have a victor.

Yes, they have to have a victor. Without a victor, the whole thing would blow up in the Gamemakers’ faces. They’d have failed the Capitol. Might possibly even be executed, slowly and painfully while the cameras broadcast it to every screen in the country.

If Peeta and I were both to die, or they thought we were…

My fingers fumble with the pouch on my belt, freeing it. Peeta sees it and his hand clamps on my wrist. “No, I won’t let you.”

“Trust me,” I whisper. He holds my gaze for a long moment then lets me go.

In The Hunger Games how does Katniss develop from the beginning to the end of the book?

Specifically in the novel The Hunger Games, and not including the other two books in the series, Katniss Everdeen's character changes through the influence of her experiences in the Capitol and in the arena of the Hunger Games.  From the very beginning of the novel, she reveals herself as being a character of great heart and compassion for the ones she loves, like when she volunteers to take Prim's place at the reaping. 

As the novel progresses, she becomes even more focused on surviving for Prim's sake, but the challenges of the Games force her choose between survival and her own humanity.  From Rue's death to the horror of the muttations, Katniss has begun to despise the Capitol, but also has the strength to challenge their regime.  By the end of the novel, the reader sees a Katniss who has matured even more and has finally realized the true dangerous nature of the Capitol.

How does Katniss Everdeen change over time?

As the story begins, Katniss Everdeen, as portrayed by her first person narrative voice, is confident yet humble, brave yet somewhat foolhardy, honest and impulsive. She is aware that she may be chosen to participate in the Hunger Games, and she prepares for this by practicing her archery skills when she hunts for food. When her younger sister Primrose is chosen, Katniss and her mother are shocked because Prim is so young, and also worried, because Prim does not have the physical prowess or assertive character necessary to prevail in the Games. Katniss impulsively volunteers to take her sister's place. This act of bravery and self-sacrifice is perhaps the first step that puts Katniss on the road to becoming a hero of the people.

But we learn that the Hunger Games, far from being a game of chance and skill, are heavily manipulated by the government. Katniss eventually begins to understand this manipulation, and her behavior and personality adapt to help her meet the challenges she faces. She becomes less impulsive, as she understands that every move she makes and every word she speaks are under surveillance. This also forces her to become more calculating, and she learns she has to sometimes resort to lies and subterfuge. By manipulating those who are observing her, she finds ways to gain the support of the citizens and to play along with the government plot to groom her to be a hero and martyr.