Which organisms would share more characteristics in common, those in the same class or the same family?
The answer to this question is directly related to how organisms are classified. I will be using the Linnaean taxonomy, which is a particular form of biological classification and I believe is the one assumed by your question.
Generally, Linnaean taxonomy includes three kingdoms divided into classes. They, in turn, are divided into orders, families, genera and species. These are called ranks and are ordered in levels:
To answer your question, we don't need to know the specific details of each rank in this classification, only the criteria used for putting animals into the same rank. And, to our luck, they are ordered and separated in a way so that as we go to ranks of higher levels (from 1 to 7 for example), we find animals that share more characteristics with each other than the ones in the previous rank.
As the family rank is two levels above the class rank, we find that animals of the same family will have more characteristics in common than the ones in the same class!
I hope I was able to give you a good explanation!