George was clearly affected by Carlson's shooting of Candy's dog, and how Candy himself wished that he would be put out of his misery when he ceased to be useful -
I wisht somebody'd shoot me. But they won't do nothing like that.
George was aware that he needed to take responsibility for Lennie, as he always had, to do the best for him. Lennie would have either died a slow death - Curley was planning to 'shoot him in the guts', been sentenced to death for the murder of Curley's wife or incarcerated in a lunatic asylum. George knew that Lennie wanted freedom and so shoots him with the vision of their farm clearly in Lennie's mind. George knows he has done the best for Lennie and Slim affirms this for him -
Never you mind. A guy got to sometimes.
George and Lenny in the story "Of Mice and Men" have been together since they were young. George has spent his life caring for Lenny. He had at one time mistreated him which he admitted, but he does genuinely care/love Lenny.
To look at reasons to make his killing appropriate there are several reasons:
1. George's shooting Lenny prevented Curley from torturing Lenny.
2. George was able to give Lenny a good death. By this I mean Lenny was not afraid, but died with the idea that things were going to be alright.
3. George did not kill Lenny on account of contempt or anger, but because he cared about him.
4. It was legal for him to end Lenny's life under the circumstances. Law was often taken into men's own hands at the time in justification for murder.
5. Lenny would have been found guilty of murder even if he had gone to a court of law.
In George's view if you have a good reason for doing it it is okay. Mercy killing has a purpose so he can justify that. He reassured Lenny things would be okay and then did to him what would inevitably happen when they were found by Curley and the men
george thought its better that he kills lennie sillently without makeing lennie feel pain or that he's being killed than a stranger killing him and letting him feel abused. :D