A mystery can be played realistically or broadly, in serious earnestness or bordering on farce. The style of performing you choose will depend on both the script, and how you wish the mystery to be perceived.
A complex, realistic mystery, can be played rather straight, with your characters indistinguishable from, and, in fact, posing as, members of the audience. The interest in a mystery of this type is directed exclusively at solving the puzzle, so this style would appeal mostly to hard core mystery fans.
In a general audience, however, you are likely to find a mix of people, some very interested in solving the mystery, and others who are just along for the ride. To keep everybody entertained, there must be something more.
A mystery/comedy format works well to hold everyone's interest -- if you can't keep 'em guessing, keep 'em laughing. Outrageous, larger than life characters call for a broader acting style, creating a production that is both a mystery and a parody of the mystery play at the same time.
The actors must be sure that they are not playing funny, however, but must seem to take their situation with utmost seriousness. It is from the exaggeration of attitudes that the humor will spring. To force laughter is to kill it.
Bringing the audience into the playing space is the first step in making them part of the show instead of passive spectators. The next step is to include them in the proceedings as if they all were characters in the play through asides, directly delivered speeches, and one on one interaction.
Whenever you direct a question to a member of the audience they may answer it, so you have to be ready with alternative reactions, and a ready wit. Anticipate possible answers beforehand so you can be ready with a comeback no matter what they throw at you. Training in improvisational comedy can come in handy here, and so can experience as a stand-up comic.
But the best defense is preparation -- you have to know your character inside and out if you are to stay in character while going off script. This means knowing your lines so well you can break the flow of the script and still come back exactly to the point you left off without struggling for lines or to remember your place.
Next Step: Characters: Status and Attitude.