You could say that it was a combination of personal and business. In the passage from the novel Pete referenced, Michael said that Vito's taking everything personally ("He knows every feather falling from the tail of every swallow...and no accidents happen to people who take everything personally," or something like that) was the key to his greatness, and thus his effectiveness.

In the movie, I believe Michael's decision was personal and business. Personal maybe partly because he wanted to avenge himself on Mac, but more, I think, because he had a personal epiphany in the hospital: When he said, "I'm with you now, Pop," I (and many others here) believe he meant more than just physical proximity to his father. He was deciding to become an integral part of the family business.