Determine why the original peach tree died. Trees that succumbed to old age or frost damage are not a problem, but you will have to treat the soil before replanting if the tree was killed by a disease or insects. A professional arborist or nurseryman should be able to help you identify the problem if you are unsure.
Remove the dead tree, using a hatchet or chainsaw to cut the tree down depending on its size. Choose the direction you want the tree to fall and cut a notch in that side of the tree. This notch should go one-third of the way into the tree’s trunk. Start at the top of the intended notch and cut into the tree at a 60-degree angle. Then make a horizontal cut into the trunk and remove the wedge-shaped piece. Walk to the opposite side of the tree and cut horizontally through the trunk two inches above the bottom of the wedge you removed on the other side. Stop cutting as the tree begins to fall and there are still a few inches of trunk left to create a “hinge” on which the tree will fall. When the tree is safely down, cut through the hinge, leaving a stump. For very large trees, it's safer to hire a professional tree company.
Dig the stump out of the ground with a shovel if it is small enough, cutting through any roots you encounter along the way and removing as many of them as possible. Rent a stump grinder or hire a professional service to tackle large stumps, grinding as deeply as necessary to remove the entire stump and as much of the old tree’s root system as possible. This depth will vary depending upon the age and size of the tree removed. Cut the stump down as close to the ground as you can before grinding to save time and make the job a bit easier.
Remove as many of the wood chips as you can during and after the stump removal process. Save the wood chips to use as mulch if your old peach tree died of natural causes, but destroy or dispose of them if the tree died of insect damage or infection.
Have a soil test done at the site. Even if the soil was rich and fertile when you planted the first tree, it may now be depleted. Work fresh, clean topsoil into the area along with any amendments, fungicide or insecticide treatments that may be necessary to treat whatever killed your old tree. Wait as long as is recommended by the treatment manufacturer to give the treatment time to work before replanting.
Plant the new peach tree as deeply as it was planted in the nursery container. If your old tree was the victim of a disease or a certain insect pest, choose a replacement variety that is resistant to the problem.