My arguments that the death penalty is cruel and unusual punishment are essentially those of Breyer in his dissent in
Glossip v. Gross. See posts #1 and #2 here:
Is the Death Penalty Constitutional?
Other than Breyer's arguments, and the facts and figures noted in those two posts, I would just note a couple of other factors that argue against (or might argue against) the death penalty. One is that redemption, rehabilitation, reformation, or whatever you wish to call it, does occur among prisoners and otherwise people who have committed heinous crimes. For instance, there is a case from the 1950s or 1960s where a man raped two high school girls, tormented a couple of other guys, and shot and killed 1 or 2 police officers, all within the span of a few hours. He was only identified and arrested some 40 years later. That was apparently his only crime; he had become an upstanding and well-respected member of his small community in another state, was a retired grandfather who did lots of volunteer work when he was arrested, tried and convicted.
Convicted murderers occasionally turn out to be exemplary prisoners who do good works in prison--e.g., teach illiterate prisoners to read.
The death penalty prevents such redemption/rehabilitation/reformation.
Another factor that might argue against the death penalty is that it a good many of those given life sentences without possibility of parole would prefer to die rather than spend 60 years in prison. And there is indeed a high suicide rate among such prisoners.