Alpine strawberries (Fragaria vesca) are a real delicacy: they are not easy to harvest mechanically or transport long distances, but the tiny berries are said to have a flavour that makes ordinary strawberries pale in comparison. They also produce no runners, can be grown from seed, and will grow in partial shade, making them easier to manage for home gardeners.
The first step in growing alpine strawberries is locating a source for seeds. Some mail-order companies carry them (try William Dam Seeds or Richters Herbs), and I recently lucked into a packet on a Thompson & Morgan rack in Canadian Tire. Most varieties are supposed to be pretty similar; 'Mignonette' is said to have the best flavour, 'Alexandria' is supposed to have somewhat larger berries with better yields, and several varieties have white or yellow berries that are supposed to fool thieving birds. 'Temptation' is a seed-grown strawberry that is frequently available, but I don't think it is a true alpine strawberry.
Sources differ on the best way to germinate alpine strawberry seeds, maybe because their usual germination rate is often no better than 60%. Some people chill the seeds before planting by placing them in a freezer for 3-4 weeks, others skip this step. I would follow the instructions included with the seeds for this. Seeds should be started very early - about 10 weeks before your local last frost date - but this gives you a good chance at harvesting some berries the first year. Sow the seeds indoors either on top of the soil or only lightly covered; they are very tiny. Germination may take up to a month, so be patient! Keep the soil moist at all times; placing your pots in a plastic bag can help with this.
After hardening them off, put the plants in the garden on your last frost date. Plants should probably be spaced about a foot apart for best production, and kept consistently moist. Alpine strawberries are perennials, and sources usually list them as hardy to zone 5, although some people claim they are hardy to zone 3. I think I've heard of people growing them here in Edmonton, but that might be wishful thinking. Northern gardeners should probably mulch them through the winter and save seeds in case they need to start again next year.