Well, you do see killing, warfare and 'hate' in species of ants, primates, wolves and incidentally in solitary predators.
I do think that you can logically make the case for caring though, as well as for morality.
Most cultures have some sort of 'treat others like you would want others to treat you' moral paradigm.
If people where to treat eachother like that, on the condition that the other person would reciprocate this strategy, then what you have is a nash-equilibrium: everyone's strategy is the best (in terms of personal gains) answer to everyone else's strategy. In this particular case it would mean that there is no other scenario where the collective outcome in terms of individual personal gains, is better. This is, of all possible worlds, the one that is by far the best for everybody. Yet, the situation is inherently stable, so it's a very sustainable situation as well.
There is Always the risk of free-riders though. Someone could try to devellop a strategy to beat the moral strategy, by simply pretending to use the moral strategy, but to 'cheat' when the situation allows for it. That's where diversity in strategies can come in handy.
And it's also why moral behaviour should not be fixated. It's just like with genetic's: ever evolving, hybrid strategy's are less likely to fall victim to all kinds of bugs.
As caring itself is a very basic behavioural mode, part of parenting, it's very plausible that in a primitive species, even a slight genetic mutation would automatically lead to a preference for caring sort of behaviour, wich would be enough to automatically result in a moral nash-equilibrium.
However, revenge is also a part of this strategy. And it is typical that very brutal and seemingly unnessecary agression is something that's only seen in highly social animals like chimpansee's, gorrilla's...and humans.