So first, about the victim. Capital punishment does not seek justice, it seeks revenge. This is an important distinguishment, sure many people conflate the two, but they are not the same at all. Capital punishment cannot deter crimes from happening, furthermore it justifies killing as a reasonable way to dispose of alleged monsters. Let us not forget that every time someone goes on a killing spree or a kills repeatedly, they believe that they are eliminating trash from society. Their self-righteousness is identical to the self-righteousness espoused in the death penalty, only their beliefs have not passed the sloppy test of democracy. People have so little faith in the justice system, but I suggest you go out and look at the various organizations that seek to support the victims and see how many of the members are lawyers and judges. These people are far more concerned about preserving victims rights than the average citizen, but as we all know, the media only ever reports the most sensational sentences and never reports the appeal courts strengthening the sentence, which happens quite repeatedly.
There is far more reason to seek to eliminate crime than to punish crime, but so many people get caught up in the punishment aspect that they forget how little punishment acts as a deterrance. Conclusively, the death penalty has no impact on the crime rate and only seeks to satisfy the most bloodthirsty of the law-abiding. I don't think that is a healthy catharsis. It reinforces the idea that violence can be solved with more violence, and that innocents dying is justifiable in the name of punishment.
On the nature of costs in the system and backlog in the courts, the death penalty can only increase this. The defense attorney of an individual facing the death penalty will stall the entire process, and apply for every appeal. They will be able to successfully bring their case before each appeal court simply on a constitutional ground, and they can tack on any issue that will guarantee a lengthy trial. Thus the death will be anything except quick and inexpensive. Also, it often eliminates judicial discretion. Consider the recent case where a father euthanised his daughter, faced with the decision of choosing acquital or death sentence for premeditated "murder", the judge might decide that justice is better served with an acquital.
Finally, on the issue of who is killed. Many of these people aren't murderers or sexual abusers, take for instance the famous Californian "three strikes" rule. Many of these people are subsistence criminals, so don't use the extremes to justify a law that penalizes the small fries.
So in summary, its far more expensive, it perverts the justice system (there is justice in this system, but thats not sensational enough to make the news) into a revenge system, it is an unhealthy catharsis/message for society (i.e. killing is an effective way to solve problems), it ignores the issue of lowering the crime rate while seeking to satisfy people about the crime rate, it does not focus solely on the most violent and dangerous offenders. From an economic standpoint, it fails by increasing costs. From a criminologic standpoint, it fails to prevent crime. From a victims standpoint, it fails for some, so it just fails. From a justice standpoint, an innocent will die, so it fails. From a sociological standpoint, it manifestly creates a more violent society, so it fails. What is left but the appeal to emotion of a few?