Strawberries propagate themselves by sending out runners that create new plants and roots. Once the new plants have established themselves, the original plant will keep extending the runners, so you'll get several new plants on each, you can cut the runner off in between the new plants, alternatively once there are a few roots you can cut the new plant off and plant it in a pot. I tend to dig up the new plants, because the shoot will just go anywhere, even the lawn. I just make new rows of plants.
Make sure you have an old fashioned variety, because hybrids won't produce runners and trying to grow from seed will also not work.
You can try and plant a dried-out strawberry, otherwise separate seeds from the fruit (they are on the outside) by blending in a blender with water and picking the seeds out of the pulp. Before you plant out the seeds you may need to keep them in the freezer in an airtight container for a while to wake the seeds out of dormancy.
Apparently due to excessive cultivating over the years of the common garden strawberry plants grown from seeds are unlikely to be similar to the strawberry they came from and more like a previous generation.
Strawberry plants keep bearing fruit for a few years but slowly diminish, which is why many people prefer them as annuals. I have noticed that the biggest fruit and the most prolific harvest takes place in the second year.