I have heard that if a Muslim kills kafirs without any Shariah reason he cannot be given a capital punishment but if a Muslim kills another Muslim or if a kafir kills a Muslim then he should be given capital punishment. To support this argument they quote a Hadees the Prophet (peace and blessings be upon him) said: “A Muslim may not be killed against an unbeliever” Bukhârî through the narration of `Hazarat Ali. However, I understand that the Hanafî law code permits retribution for the non-Muslim victim’s family to be administered on that family’s behalf against a Muslim murderer. The evidence for that is "And we prescribed for them therein (in the book): The life for the life, and the eye for the eye, and the nose for the nose, and the ear for the ear, and the tooth for the tooth, and for wounds retaliation. But who so forgives it (in the way of charity) it shall be expiation for him. Whoso Judge not by that which Allah has revealed: such are wrong-doers." (Sura Al-Mai'da 45) “O you who believe! Al-Qisaas (the Law of Equality in punishment) is prescribed for you in case of murder: the free for the free, the slave for the slave, and the female for the female. But if the killer is forgiven by the brother (or the relatives) of the killed against blood money, then adhering to it with fairness and payment of the blood money to the heir should be made in fairness. This is alleviation and a mercy from your Lord. So after this whoever transgresses the limits, he shall have a painful torment” [al-Baqarah 2:178] Prophet (peace and blessings of Allaah be upon him) said: “It is not permissible (to shed) the blood of a Muslim who bears witness that there is no god but Allaah and that I am the Messenger of Allaah, except in three cases: a married adulterer, a soul for a soul, or one who leaves the religion and splits from the jamaa’ah.” al-Bukhaari; Muslim). Can you please explain with the evidence from Quran and Sunnah which is a right opinion?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

First of all, you should know that a Muslim should not be killed for killing a belligerent non-Muslim according to the consensus of the scholars . According to the view of the majority of the scholars a Muslim should not be killed against a free non-Muslim under the Muslim rule. The evidence about this is the saying of the Prophet : "A Muslim should not be killed for killing a non-Muslim." [At-Tirmithi]

Moreover, according to the view of the majority of the scholars the title (and rulings) “disbeliever” is applicable to a free non-Muslim under the Muslim rule. However, Abu Haneefah, and the scholars of his School of jurisprudence are of the view that a Muslim should be killed for killing a free non-Muslim under Muslim rule; their evidence is the two verses which the questioner mentioned. Nonetheless, the correct opinion is that of the majority of the scholars that is based on the above Prophetic narration, which is a direct proof related to the case of dispute. 

Allaah Knows best.