Persimmon seeds have a bad habit of rotting rather then germinating!
If you are determined to try growing them from seed, sow them in good quality loose soil-less mix! Use 4" plastic pots, but sterilize them first! Seeds should be just covered with soil (barely moisten the soil, you want it damp but not wet). Place pots in zip-lock bags and place in the second shelf in your fridge and in the front rather then the back (the back of the fridge tends to be too cold as is the bottom shelf). Leave them there for a couple of months. Then bring them out to let them germinate!
If you don't like the idea of using the fridge, do the same, but place the bagged pot, or pots, outdoors in a sheltered shaded spot when you can keep an eye on them come late winter/early spring.
To cut the time frame from germination to flowering size tree, you could always go hunting for seedlings from the wild ones you see around your area! Now would be a good time to go hunting since they should be going into winter dormancy! Try to dig out as much of the root system as you can!