Is capital punishment just state-sponsored murder, petty revenge, an example of hypocrisy in that we say "don't do this" and emphasize the point by doing it ourselves? There's certainly something to that, but it's also a vast oversimplification.
There are no absolute, situation-independent, context-free rules. Not even "thou shalt not kill" -- consider living wills and doctor-assisted suicide for counterexamples, as well as the bigger one of "that man is about to press a button which will blow up a billion people and we can't stop him any way but by killing him" (maybe you wouldn't pull the trigger, but I would, and sleep fine that night). So it's making things way too simple to say that we can't say "you're not allowed to murder" (where "murder" is a much more specific thing than "cause a person's life to end") means we can't ever have cause to end someone's life.
The question is, does capital punishment serve a purpose great enough to justify it? Clearly we can't be killing people merely for the satisfaction of seeing that a bad person got his comeuppance; that's mere vengeance. What, then, is the real purpose of capital punishment?
The best answer is deterrence. If people know that the stakes are as high as death, they are, it is theorized, less likely to commit certain crimes. Research about this factor is extensive but often inferential -- a particular crime is less common in one jurisdiction with capital punishment than another without, but that doesn't mean it's not because of many other differences, or that if capital punishment were used more consistently the crime wouldn't even out (assuming people are going to do it with or without the punishment, but just bring it to where the penalties are less). Anyone who wants to argue either side can certainly bring evidence and reasoning to bear. I think on balance the preponderance of evidence suggests there is a non-trivial deterrence factor. But so long as we can't really say with certainty how effective a deterrent it is, it's hard to argue whether it's sufficient to be worth the cost.
The trouble is, you can't really do systematic, exhaustive research on this. You can't create a realistic but artificial situation, where you can control for other factors, due to both ethical and practicality concerns. So all you can do is infer from lots of real-world data and try to gather enough to eliminate for the other factors like cultural differences. Trouble is, even if you can come up with something pretty statistically certain, you'll never convince the average person with it. Data like that is unconvincing to most people because they don't understand the statistical methods used, so they cling to anecdotal cases where those techniques failed, and don't trust them.
However, I think we can safely say that those who claim that, by "sending the wrong message," capital punishment can encourage murder, have no real evidence behind this claim. That's just the kind of specious reasoning that posits that all killing is equal and then concludes that all killing is equal, which was already discussed.
Until there's such a heavy preponderance of statistical data and analysis that it weighs so firmly on one side or the other of the deterrence question that we have to agree with it, we're never going to get to a point where we can be sure. That's why I don't take a firm stance either way. But even when that day comes, the majority of people arguing will be arguing based on irrelevant oversimplifications, so it won't matter to them. The inferential nature of the research will simply give them the thin end of a wedge that's sufficient for them to brush aside the conclusions no matter how certain they are, allowing them to go back to their oversimplifying analogies.
Pity, too. Some of the people most prone to that are the same people who have the most energy that they could be using to make the world better, if they weren't wasting it pursuing whichever side of the illogical counterarguments happen to have caught them first.