As with most fruit trees, growing an apricot tree from seed requires a period of cold storage, also known as afterripening, to simulate the normal dormancy of winter conditions, before the fruit will sprout. You can begin sprouting your apricot seedlings indoors by afterripening them in a cool, dark place, or you can simply plant them as in the soil in the fall and allow them to afterripen on their own in the natural climate.
Remove the apricot seed from a healthy apricot, rinse away all remaining apricot flesh, and leave the seed out on a paper towel in a room with good air circulation until it's dry.
Place the apricot seed in a tightly sealed container and set aside in a dark, cool place, until mid-January.
Remove the seed from the container in mid-January. Lightly moisten peat moss, sand or shredded paper towels with water, then add enough of this mixture to fully cover the apricot seed on all sides. Place the seed in this mixture, making sure it is fully covered.
Place the lid back on the container and put the container in a dark environment between 40 and 50 degrees Fahrenheit for between sixty and seventy days. You haven't actually planted the seed yet, so it doesn't need sun; you're just getting it ready for planting.
Remove the seed from its cold storage place in early April. Prepare a seedbed--this can be done in a container as well, ideally at least a foot deep--by filling it with garden soil and digging a furrow that's twice as deep as the longest dimension of the apricot seed. Place the seed in the open furrow, cover it with a light covering of soil, and add an inch of sand on top of this.
Keep the soil around your seed moist but not wet until it germinates. There is no need to fertilize until your seedlings have sprouted and are between six and eight inches tall.