The trait that motivated Kouga, the one thing that kept him going, was loyalty to the pack. His pack.
Loyalty, the reason why he hunted Naraku. How else would the souls of his deceased companions be able to rest in peace? How else would those that remained be able to sleep easy?
Loyalty was something that worked both ways. The loyalty Ginta and Hakkaku showed in following Kouga was repaid in the way Kouga would lead the way. Kouga would always watch out for danger and choose the safest possible route. If he could, Kouga would avoid danger, just to keep those two safe. If danger could not be avoided, Kouga would fight to protect them - and they vice versa.
Loyalty was also a reason why Kouga despised Inuyasha. What did dogs understand of loyalty?
Sure, there are those that say, dogs are famed for their loyalty. But they are not. Dogs have a sense of loyalty that wolves find warped. Dogs can easily be swayed to suit their master. Dogs can be moulded into mocking forms, pip-squeak or brute savage. Hunting down their own kind... for the whim of their human. Their loyalty is for individuals. Not the pack.
It was loyalty that changed Kagome from a prize for his tribe to a desirable woman in her own right. She had loyalty to her pack. And her pack clearly loved her for it.
It wasn't a family-based pack, but nonetheless, it was clearly her own. She adopted the members and they gave her love. She was their heart and soul - but that didn't prevent the way she easily offered her loyalty, helping where she could. The way she transformed from mild-mannered to powerful fury in order to defend and protect her pack.
Kouga wanted to be a part of that loyalty.