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Is he or is he not? Does the society accept him/her like that? This question came up to my mind when my father mentioned about a professional killer whose nickname was The Iceman.

PS. Don't know what about the tags.. could you please help me with them.
Thank you in advance.

'Good man' is a comprehensive, all things considered judgement or assessment. I'm not sure that this is possible on the slender data supplied.

Suppose the doctor saved a great many lives in his professional capacity, but only because as a member of a team he could not avoid doing so (he would much rather have let the patients die or actually killed them) and his murders inflict suffering for his/ her personal pleasure. This is hardly the picture of a good man.

But the situation could be different. The doctor might be totally dedicated to saving the lives of patients wherever possible and murder purely when he morally considered mercy killing to be justified or even required in the interests of alleviating suffering. It would not be totally implausible to describe such a doctor as a good man. (He might have an erroneous conscience in the case of mercy killing but that bare fact would not make him a bad man. There is moral virtue in following one's conscience.)

So I think the judgement swings one way or another depending on the circumstances : 'circumstances alter cases'. We would need more data about the doctor than the question provides.

Calling such a person "good" would be quite a stretch, even for a utilitarian. If all the people he kills are murderers themselves, then one could argue that he's helping society. (However, there are a number of arguments against such a judgment.)

However, the great majority of doctors save lives and do not murder people. In fact, the medical profession has a pretty strict code of ethics which ought to make murder even more unpalatable. I think most doctors would therefore call this guy a "bad doctor."

In fact, if his crimes were disclosed, he would likely lose his license and would almost certainly wind up in prison.

Saving lives might be a good trait, but if this guy was in prison, could his patients not be saved by other doctors?

Verdict: This doctor is bad.

Let's say this man saved 100 lives as a doctor, and murdered three people. It would be tempting to say "society is 97 lives ahead, so he is a good man."

But that is wrong. If he didn't work as a doctor, someone else would have done his job, and those 100 lives would have been saved anyway. And the doctor is paid to save lives, the taxpayers that fund him have just as much to do with the saving of lives as he does. He doesn't save lives to be a good person, he saves them because that's his job. I know a car mechanic who saved the lives of hundreds of cars. There is no reason why you would think the doctor is a better person than the car mechanic.

So no, he is just evil and nothing else.

Society's judgements/reasonings about such matters can be goofy, whereby your question's relative merits/demerits can't really be quantitatively assessed to arrive at an unambiguous conclusion. Not directly related to your question, but addressing the goofiness contributing to its unaddressability, is the following (apparently true) story...

On Oct 13, 1943, poet and Jehovah's Witness Robert Lowell was
sentenced to a year in New York's West Street jail for evading
the draft. There he met Murder Inc's Louis ("Czar") Lepke, who
mentioned, "I'm in jail for killing somebody, why are you here?"
Lowell replied, "I'm in jail for refusing to kill anybody."