If 'an eye for an eye' is how the Hebrew people delineated the law of retaliation, is going beyond the law of proportional response with respect to killing then a breaking of the sixth commandment, in that disproportionate killing becomes murder?
The usage of "eye for an eye" is part of a judicial response to an action. The Torah in the pasuk is speaking of the judicial determination of punishment for damages and the assessment of the monetary fine. The death penalty is limited to those cases which are specifically subject to the death penalty. Since the death penalty is given by the court, it is not considered murder.
There is no evidence anywhere, literary or archaeological, that a literal "eye for an eye" was Jewish practice at any time. Nor is there the slightest hint in the Talmud, the principle body of Jewish law that this verse was ever taken literally. It is simply an erroneous assumption based on the literal reading of the verse.
The Talmud, in fact, records a lengthy discussion of this verse, (Bava Kama 83b-84a). The Talmudic sages bring a number of compelling proofs, both logically and from the inference of other verses, showing one should not even entertain the thought that "an eye for an eye" is to be taken literally. Maimonides, the renowned 12th century sage, further cites the verses in Exodus 21:18-19 which openly speak of damages in terms of monetary payment. Hence, a few verses later when the Torah speaks of "an eye for and eye…" it is obviously referring to the same sort of payment. Other early sages bring additional proof: if literal, if the perpetrator injures another and minimizes his sight by one third or half, how is it possible to do the same in punishment, no more and no less?
The key principal is that the Torah cannot, and was not meant to be understood literally. Only with the Oral Tradition given together with the written can the Torah be understood correctly and accurately.
Similarly, the extrajudicial death penalty given by a king is also not muer when done under the specific circumstances and methodologies as set forth in the Oral law (such as for rebellion against the king).
Bais Din also has certain extrajudicial powers as shown in Choshen Mishpat 2:1
Every beit din, even if they weren't ordained in the land of Israel, if they see that the nation is engulfed by sins, (and that it is a need of the time) (Tur), they can judge both [life-and-]death and monetary [cases] and all matters of punishments, and even if there isn't complete testimony. And if he refuses to listen (lit. is mute), they let him be handed over to idol worshipers. (And they have the power to make his money ownerless and to destroy it according to what they see (to guard against the rebelliousness of the generation) (Tur in the name of the Rambam, Sanhedrin chapter 24). And all of the [court's] actions should be for the sake of Heaven. And [this allowance] applies specifically to a leader of the generation or good people of the city that are under the beit din's supervision. Note: And similarly, we act like this in every place, that the good people of the city in their city are like the highest beit din, lashing and punishing, and their [decision that something is] ownerless [is in effect and it is considered] ownerless, even though there are those who argue and reason that the good people of the city don't have this power, but only to enlighten the congregation about what were the traditions from oldest times or what [traditions] were accepted by everyone, but they are not permitted to change a matter [whose changing would be] beneficial to some and causing loss to some or that would ? money that is not accepted by everyone (Mordechai, last Perek "HaGozel") — nevertheless, they go according to the custom of their city, and all the more so if they had accepted anything on themselves, so it appears to me (and see Yoreh Deah section 228 for laws of decrees and stringencies of a congregation). The later authorities have written in their responsa that someone who is obligated for lashes should give 40 gold [coins] in place of the lashes (Mahariz section 147 and Maharam of Rizbork), and they didn't say the ruling, but rather ruled like this because of the [needs of the] time, but it is in the power of the beit din to lash him or take money according to what [the need] seems to be in their eyes, depending on the matter, beyond the fence of the law (and see later at the beginning of section 425 in the note).