The scene where Richard kills the kitten serves mostly to introduce the conflict between characters, Richard and his father. Trying to sleep so he can rest up for his job as a night porter, his father yells to the boys to kill the noisy kitten. This, of course, is not taken to be literally but Richard's act of defiance is precisely in this act of failing or being unwilling to interpret his father's words. He kills the kitten.
The situation is really amoral, neither good nor bad in itself because on the one hand Richard followed his father's command, but on the other hand he killed. The scene illustrates the conflict between characters in power and Richard himself, a tension that will follow him throughout the narrative. The death of the kitten also symbolizes the realm of the unspoken and the failure to look for deeper meaning.