Most tree seeds remain in a dormant state that keeps them from sprouting, according to Richard Jauron of the Department of Horticulture at Iowa State University. You can use stratification or scarification to break the dormancy. Scarification is breaking through the exterior coating--in this case the plum's woody pit--and stratification puts the seed through a period of cold. The plum seed needs about 90 days of stratification before it will germinate.
Break open the hard woody coating on the plum pit. You can do this with any number of implements, such as a hammer or a nut cracker. It tends to be hard wood, so be careful to just crack it open and not smash the tender seed inside. This action will provide the scarification necessary for the plum seed to sprout.
Place the moistened mixture of sand and peat moss into the coffee can. It should be damp but not wet or the seed will mold and rot. The seed should be buried at least 2 inches down for proper coverage. Cover the can with the plastic lid.
Store the can in the refrigerator--maybe in the back of the vegetable drawer to keep it out of the way. It will need to stay there for 90 days. Start checking the can for sprouting seeds after the 90 days. As soon as the sprouts start, remove the can from the refrigerator.
Prepare your plant pot with regular potting soil. Dig a 2-inch deep hole in the soil and gently place the sprouted seed in place. Cover it with the soil and tap it gently to make sure the dirt is touching the seed. After a week or so, you should see green shoots emerging from the soil. Keep it in a sunny window and water it weekly until the soil has warmed to around 50 degrees outside. Transplant it and treat it as a normal plum tree seedling.