"Business or personal" always seemed like a false dichotomy to me. They're always intertwined.

He steered his business interests to follow his personal desire for (his vision of) legitimacy. But it still was good business.

He killed his brother out of anger ("He injured me."). But that probably helped him in business, burnishing his fearsome reputation.

The same applies here. He was angry and terrified that his father might be murdered. He wanted to get McCluskey back for breaking his jaw. So, in that sense it was personal. But it was also the best move strategically. They did need to kill the Turk and there seemed to be no other way than for Michael to do it himself.

To me, Michael was successful because he was able to use his personal motivations to improve his business.