Dr. Valentine is a surgeon, so his "business"—what he does for a living—is or includes performing surgery. Surgical procedures are not without risk; if the procedure goes wrong, the patient can even die of the consequences.
This is how a surgeon may unintentionally cause somebody's death. Cynically put, one may say that the surgeon's "business" or job has killed the patient. This is what Dr. Valentine's wife meant when she said that "[her] husband isn't killing anybody except in the way of business": his work may unintentionally cause a patient's death.