Could the pursuit of happiness be considered the purpose of all life? Is it not what all life strives for?

It really depends upon what you mean by "happiness." If you mean the somewhat fleeting and temporary experience (like the "happy" in "happy hour"), I think a life aimed at happiness would turn out to be fairly meaningless and empty. But if by "happiness" you mean something like "flourishing as a human being" or "having a complete and full life" or something like this, then it does seem like a reasonable overall aim in life.

Philosophers who think that this aim can be one's guiding aim in life (and the aim from which all value in a human life flows) are called "eudaimonists," from the ancient Greek word, eudaimonia. Most ancient Greek philosophers thought that eudaimonia (which is generally translated as "happiness") was and should be the guiding principle and the ultimate aim of human life. If you would like to see a fine example of how a serious ethical work takes this position, I would enthusiastically recommend Aristotle's Nicomachean Ethics, especially when he lays out his eudaimonistic foundations in Book I.

Of course, other philosophers have also taken happiness seriously as an end of life and action. A very different approach that also continues to have wide influence in ethics, which also takes happiness as a foundational value, may be found in John Stuart Mill's Utilitarianism.

Read another response by Nicholas D. Smith
Read another response about Happiness, Value