sonny and michael's "business not personal" exchange in the movie is often misunderstood.
people seem to think that this means that you must emotionally distance yourself whenever you do a hit, in order for it to be "professional." this is not the case.
what michael and sonny are talking about is planning a specific hit. even when you hate someone as much as Michael hates McClusky and the Turk, when you plan out a hit you must be stone cold professional. the decision must make sense from a business standpoint even if the hit itself is laced with vendetta or emotion
obviously Mike has personal feelings but if hitting the Turk would have hurt the family, or if they would have had to wait years to get the Turk for the sake of the family business... then Mike would have been willing to do that... as a business decision.
"it's business not personal" has to do with how you plan and carry out a hit. obviously many of the hits in Godfather are personally motivated but only executed when it makes sense from a business standpoint.
a good comparison is how Michael and Sonny handled Carlo. This rat's been beating on their sister and obviously both hate this bastard equally. however, Sonny takes this personal and acts rashly and ends up on dead on the freeway. for mike however, it's a business decision to bide his time, keeping Carlo close and getting all the 411 he needs from him before he arranges to wack him.