First of all, you need to pick up the dead body if it lies on ground or take it out if it is died in the cage. Make sure you do not get your hand hurt with the bars of the cage.
Now, you should take a box making sure that the size of the box is appropriate for the dead bird. You can use check book box, shoe box or cereal box for this purpose. But keep in mind that it should be large enough that does not require you to stuff the dead bird in it.
The next important step is finding the right place where you want to bury the dead bird. You can find a corner especially in your back yard or beside the bush. You can also bury it besides a big tree that has grown in your yard.
After selecting a place, you need to dig a hole which should not be less than two feet. Make it wide enough to accommodate the box of dead bird easily.
Put the box in this hole and refill it with the soil or dirt that you have dug out for the hole. Make sure you have levelled the surface and sprinkle water on the surface in order to keep the surface smooth.
Place a stone or other mark in order to identify the place where you have buried the dead body of your pet bird. You can also use a board stating the name that you selected for your pet bird.
If your bird has died due to a disease then you need to be careful in touching it. Always avoid touching it with bare hands and use disposable gloves which will protect you from catching germs from the bird.