How has the character of Bruno, the protagonist, evolved from the start to the end of the story?

Bruno’s character does not significantly evolve from the beginning to the end of the story. He is introduced as and remains a kind-hearted, naïve, innocent child who follows his own unwitting logic based on his limited and sheltered life.
In the beginning, Bruno is presented as a nine-year-old boy living in Berlin during World War II—a time and place most people would choose not to be. However, given his young age and his father’s rank as a Nazi officer, Bruno has no choice in being caught in the war. Bruno's naiveté is evident in his lack of proper pronunciation of simple words, as well as in his befriending an “enemy,” Shmuel. This friendship further illustrates Bruno’s sheltered life as a child of privilege, since he doesn’t even grasp the fact that the Nazis, of whom his father is a member, are trying to eradicate Shmuel’s people.
Throughout the novel, different people in Bruno’s life keep sheltering him and urging him to keep his head down and wait for the storm of war to pass, as if the war should not affect him. In chapter 6, Maria says to him:

Bruno, if you have any sense at all, you will stay quiet and concentrate on your schoolwork and do whatever your father tells you. We must all just keep ourselves safe until this is all over. That’s what I intend to do anyway. What more can we do than that after all? It’s not up to us to change things.

Bruno seems to accept this advice, since, even though he is interested in a wide array of subjects—including art and exploring—he still wants to be a soldier just like his father when he grows up.
In chapter 13, when Shmuel tells Bruno that “There aren’t any good soldiers,” Bruno insists that his father is a good soldier. This shows that despite having developed a friendship with Shmuel, Bruno still doesn’t understand that as a Nazi officer, his father really isn’t a good soldier—especially not good for his new friend, Shmuel.
It is not until chapter 14 that Bruno begins to comprehend the deep differences between him and his new friend. At this point we see him begin to develop a deeper connection with Shmuel as he contemplates what is actually happening around him:

Bruno tried to return to his book, but he’d lost interest in it for now and stared out at the rain instead and wondered whether Shmuel, wherever he was, was thinking about him too and missing their conversations as much as he was.

At this late point in the novel, the shell of Bruno’s sheltered life begins to crack; he is ready to emerge and fly to the other side of the fence. But he does not understand that there can be no return from this step, which will be a certain death sentence. He simply wants to be with his sad new friend.
In chapter 16, we are face to face with the harshness of reality and the dangers of Bruno’s innocence: “ ‘I look just like you now,’ said Bruno sadly, as if this was a terrible thing to admit. ‘Only fatter,’ admitted Shmuel.”
In the end, the reader mourns the demise of Bruno and Shmuel, and we are left with the harsh reality of war and its devastating effect on children above all else. Bruno, who wanted to be a “good soldier” like his father, fails to see the truth that is before him. Instead he lives and dies with the innocence only found in the heart and mind of a child.