What actions show that Peeta was able to stay true to himself during the Hunger Games as he wished to?
Peeta began the games by seeking a way to keep both he and Katniss alive. He wanted to protect her but he certainly didn't want to just lay down and die. His first step in this direction is to team up with the career tributes. It looks like he is turning on Katniss, but really he is just trying to protect her. We see Peeta's kind nature in the lack of people he kills. He only kills one girl (aside from Foxface who died accidentally) and she was already suffering from an attack by the other career tributes. We can see this more as an act of mercy than a killing.
Peeta helps Katniss to escape into the woods with the bow and arrow after the tracker jacker attack. He even faces mortal peril for allowing her to go free. We see Peeta leave the group and go into hiding rather than seeking out Katniss or trying to kill other tributes.
Peeta tries to continue protecting Katniss. Even after she finds him and is caring for him, he still tries to protect her. We see this prominently when Katniss has to drug him in order to leave and seek the medication he needs to survive.
Peeta's character remains constant throughout the story. Even at the very end when Katniss considers turning on him, he still refuses to change into the hunter and murderer they want him to become.
Peeta has always been kind and always loved Katniss from the start. Since the beginning he made sure to keep Katniss safe, even when he was caught with the other victors who were trying to reach Katniss, he made sure he was safe, he warned her beforehand in a situation when she was close to danger. Through the entire time, he loved her and did his best to save her.
He hardly killed anyone in the first book. As far as I know.. he only killed the girl from District 8.. and Foxface indirectly.