We assume you are handling a dead mouse because: 1) you found a dead mouse in your house or your yard, or 2) there is a dead mouse in a mousetrap.
Concerns to be aware of are micro- or macro-parasites on the mouse that may be transmitted to you, contamination of items the mouse may come into contact with, and walking through or working in a mouse-infested area.
For a dead mouse outside, the easiest route is to slip one or two plastic bags over your hand (make sure there are no holes in the bag!), pick the mouse up with the “gloved” hand, turn the bag inside out with the rodent inside, and seal off by tying the ends or using a twist-tie. Dispose in the garbage.
For a dead mouse inside, use the same procedure. However, in many parts of the country, hantavirus pulmonary syndrome (HPS) is a disease of concern. The U.S. Centers for Disease Control and Prevention (CDC) has listed important steps in cleaning up trapped or dead mice (http://www.cdc.gov/rodents/cleaning_up/index.htm).
First, be sure that you are in a well-ventilated location and are not stirring up fecal- and urine-contaminated dust which can transmit the hantavirus through the air. The CDC recommends ventilating the area by opening the doors and windows for at least 30 minutes. If you have any questions, consult with a local public health official. Always wear intact rubber or plastic gloves when cleaning or disinfecting items contaminated by rodents.
You have a dead mouse in a mousetrap? You can throw the trap away with the mouse, or remove the mouse and disinfect the trap. Traps can be disinfected by soaking them in a solution of three tablespoons of bleach per gallon of water or a commercial disinfectant-containing phenol (such as Lysol). Note that chlorine will corrode the metal parts of a trap. After handling rodents, resetting traps, and cleaning contaminated objects, thoroughly wash your gloved hands in a general household disinfectant or in soap and warm water. Then remove gloves and thoroughly wash your hands with soap and warm water.
For extensive mouse infestations, see the CDC recommendations at http://www.cdc.gov/ncidod/diseases/hanta/hps/noframes/prevent4.htm. Clean-up requires special precautions and equipment.
This is a good time to conduct a check around your house to make sure you have closed off any potential openings (1/4 inch or greater in size) and reduced or eliminated potential mouse harborage around your house or in your yard.