The careful examination of a horse that has died should primarily be conducted by your vet. That said, you can gather valuable information to share with your vet, information that may also help you save your other horses.
Consider the following questions:
Do your other horses on the premises appear normal and healthy? If there is any question, briefly evaluate your other horses before you examine the dead horse.
Observe the environment and consider the following questions that your vet may ask: Do you notice anything suspicious in the environment? Has anything changed in the environment or has there been any unusual activity recently (i.e. construction, painting, application of an herbicide, etc...)?
Are there any poisonous plants that your horse could have ingested? Did you change your horse's diet in the last few days? Was your horse wormed or vaccinated in the last few days? Has the horse received any other medication in the last few days? Has there been any drastic change in weather (hot or cold) or lightning storms?
What was the horse's general health like? Their body condition (thin, obese or normal) and coat condition? Does the horse have a history of any illness? Where did the horse die? In its enclosure? Out in the open? Near feed?
If appropriate, now examine the horse more closely. Put on gloves and check their mouth for feed, other plant material, or anything else. Is there any in their mouth? If so, tell your vet. Is there blood coming from any of the orifices? Is there evidence that your horse struggled, such as bruises, swelling, wounds, areas of hair loss and scrapes, especially around the face? Do you notice any burnt spots on the skin, especially on the legs?
Try to determine how long ago the horse died. Within a few hours, rigor mortis sets in, the intestine of dead horses rapidly fills with bacterial gas, and the eyes become cloudy and shrunken. Can you still move the limbs or are they rigid? Is the abdomen very tight and drum-like? Are the horse's eye(s) clear or cloudy?
Share your findings with your vet and discuss the advantages of a post-mortem exam.
Tips for safety & Success
Obviously, this skill is not for everyone. The passing of a horse is a very emotionally traumatic event.
At minimum, try to ensure that your other horses and animals are well. Carefully consider common factors between this horse and others. Identifying common factors could help your vet save the lives of your other horses.
Do not handle dead animals without wearing gloves. If you suspect that the horse may have had a contagious disease, consider the health of your other horses. Ideally you should also wear a respiratory mask.
While it is extremely rare, some diseases can be passed to humans by direct contact with a dead horse, or breathing in of infectious agent released by a dead horse. Rabies can be transmitted to a human through the saliva and blood of a dead infected animal.
Keep your other animals away from the deceased horse as well, and cover it with a tarp, held down with bricks or rocks, until your vet can promptly examine it or until it is properly disposed of.