It is hard to use any Supreme Court cases to argue that the death penalty is cruel and unusual punishment under the 8th Amendment. If there were cases in which the Court had ruled in this way, the death penalty would not be legal today.
If you want to argue that the death penalty violates the 8th Amendment, you have to do so on the basis of evolving community standards. Clearly, the people who wrote the Bill of Rights did not think the death penalty was unconstitutional because they said, in the 5th Amendment, that the government cannot take a person’s life without the due process of law. This implies that the government can take a person’s life after giving them due process. This shows that the people who wrote the Bill of Rights had the idea that the death penalty was constitutional. However, you can argue that we, as a society, have moved past the time when we thought that it was acceptable to take a person’s life in revenge for a crime. You can argue that very few countries around the world that we would deem civilized still use the death penalty. In these ways, you can say that the death penalty was not previously cruel and unusual but that it is now.
It is much easier to argue that the death penalty is constitutional under the 8th Amendment. For example, you can look at Glossip v. Gross, decided in 2015, which says that lethal injection using a particular three-drug protocol is not cruel and unusual. As another example, you can look at Kennedy v. Louisiana (2008), in which the Court ruled that it was cruel and unusual to execute someone for a crime in which the victim did not die. This implies that the death penalty is not cruel or unusual when used as a punishment for murder.
The Supreme Court has never ruled that the death penalty itself is cruel and unusual, only the various ways it has been administered. Therefore, arguments that say that it is cruel and unusual have to be based more on moral grounds than on the use of precedent.