The only crime defined in the Constitution itself is treason. There are statutes defining “murder,” but those could be changed or repealed. There is also a common-law definition of murder, which some judges might rule applies in the absence of any statute superseding it. (This theory has applied to some other areas of law, but so far as I know, this situation has never arisen with murder.)
As others have mentioned, the Fourteenth Amendment does prohibit both the federal and state governments (and by extension, other local governments created by either) from depriving any person of life “without due process of law.” Government agents such as police, though, are allowed by law to kill people under some circumstances, such as when they reasonably feel that they’re in danger, and this is deemed due process of law.
The Fourteenth Amendment also guarantees each person equal protection of the law, which might be the basis of an argument that it’s unconstitutional to allow some people to be murdered with impunity but not others. Historically, this did not force states to do anything about lynchings, though. There is also a prohibition on “cruel and unusual punishment” that, some have argued, prohibit execution at least for some offenses where it would be disproportionate. Some activists would like the courts to declare that capital punishment or abortion are always unconstitutional, but the courts have declined to do so.
There have also been a small number of times in history—most famously, the Nuremberg Trials—where people whose government authorized them to commit genocide have been tried for crimes against humanity. There is no way to force a nuclear power to submit to such a tribunal (although this has not always stopped the International Criminal Court from issuing symbolic indictments it has no power to enforce).
