I would argue that altruism does exist, it's not just not the pure ideal of altruism that some people think it is. Altruistic behavior is our behavior that is irrational in terms of personal survival, but somehow helps the survival chances of species or individual members of the species. It doesn't matter whether the altruistic behavior makes the organism feel good or not, though typically it does and that's why they do it.
In primates (including early humans) altruism stemmed from limited inbreeding (a typical trait of social organisms). It turns out that social animals are more willing to help out fellow members of their species that share more resemblance to them. The benefits/cost of altruism aren't necessarily realized by the individual itself, it is a random mutation that favored reproduction of social species, and thus, those groups of primates had a higher chance of survival than groups with more in-group competition.
Altruism - EvoWiki