Sometime in the course of the play each character should, willingly or unintentionally, reveal their potential motive for murder. In The Dying Old Man on the Flying Trapeze, for example, we learn that the victim's wife, Esmerelda, found out that he was going to remove her from the act, his daughter, Sarasota, was tired of his running her life, The Human Cannonball wanted to eliminate his objections to marrying Sarasota, Spunky the Clown was being blackmailed, Leo the Lion Tamer wanted Esmerelda, and Nimrod the Strongman wanted to take over his act.
Greed, blackmail, lust, hate, envy, ... there are any number of motives for murder. The only limit is that the audience must believe the motive deeply felt, and sufficient cause for murder. The characters must be seen as capable of killing, either through temperament or sufficient provocation.
One particular character must be picked as the actual killer, and clues devised and planted throughout the story that point to this one person. The murderer should not be the most obvious suspect, nor necessarily, the least likely, but one that the audience will eventually accept as a possibility once their guilt is revealed.
For some reason, in every murder mystery I have participated in, spanning a wide range of audiences, the characters who have received the most guesses pinning them as the murderer are always the attractive young women. In The Dying Old Man on the Flying Trapeze, Sarasota is always the front runner. Similarly, in Politics Can Be Murder, Frances the sexy secretary comes out on top, followed by bimbette Cookie Jarre. In our other mysteries, independent of the attractive actress playing the part, it is always the same.
I don't know if this is the result of some sort of cultural bias against women, or if people assume that the women are the least likely suspects, and therefore guess them when they have no solid basis to make a selection, but the effect is there. This is why Sarasota and Cookie have never been assigned the role of murderer, but remain red herrings.
Once you have decided on the victim and killer, you will have to decide on a method. A straight forward gunshot is direct, but, keep in mind, if the murder is to be carried out in view of the audience the killer will have to be hidden, and the other suspects off stage.