How does the death penalty violate the 14th amendment?
There is an argument that the unequal application of the death penalty violates the equal protection clause of the 14th Amendment. But keep in mind that this is an open question - there is an argument that says it doesn't and so it isn't settled if the death penalty is or isn't a violation of the Amendment.
First let me address the question posed - How does the death penalty violate the 14th Amendment? - by saying this: it's an open question. It has been argued that the death penalty violates several Amendments (not just the 14th) and there are other arguments that support the death penalty.
I'll talk about the argument that argues that the death penalty violates the 14th Amendment.
First off, the 14th Amendment states:
Amendment XIV, Section I (ratified 1868):
"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws."
It's within that last line, the nor deny to any person within its jurisdiction the equal protection of the laws is where we start. It's a well known fact that the death penalty (in fact, this also extends to jail terms and sentences and all forms of judicial punishments) is handed in a discriminatory fashion, based on social class, race, and gender. And so with this particular punishment being unfairly handed out, the entire punishment is seen as a violation of the 14th Amendment.
The link below does a great job of talking about arguments, both pro and con, regarding the death penalty in relation to Amendments 5, 8, and 14.