Summarize:

Take the viable seeds and wash them thoroughly to remove any pulp or other matter.  Soak them in a small amount of distilled water for up to 24 hours. Many seeds require a period of cold, moist conditions to begin the germination process.  In nature, this is achieved when seeds sit in the ground over the winter. You can mimic these conditions through the process of stratification. For grape seeds, the best time to begin stratification is in December (the northern hemisphere winter).  Prepare a bed for your seeds. Fill an airtight bag or other capsule with a soft medium like wet paper towels or sand, vermiculite or peat moss that has been dampened. Peat moss is the best choice for grape seeds because its anti-fungal properties will help eliminate seed-damaging mold. Tuck the seeds into the bed. Cover them with about 1/2" (1.25 cm) of growing medium. Refrigerate the seeds. The ideal temperature for stratification is a steady 35-40 ºF (1-3 ºC), so the refrigerator is a good place for this process. Keep the seeds refrigerated for two to three months. Do not allow them to freeze. In early spring, remove the seeds from the refrigerator and plant them in pots filled with good soil. Plant seeds individually in small pots, or in larger pots with at least 1 1/2" (3.8 cm) between them.  Make sure your seeds stay warm enough. To properly germinate, the seeds need daytime temperatures of at least 70ºF (20ºC) and nighttime temperatures around 60ºF (15ºC). Use a greenhouse or warming mats to keep your seeds at the correct temperature. Keep the soil moist but not too wet. Mist the surface with a fine spray of water when it starts to look dry. Check for growth. Grape seeds typically take between 2-8 weeks to sprout. When your seedlings have grown to about 3" (8 cm), transplant them to 4" (10 cm) pots. For the healthiest plants, keep your seedlings indoors or in a greenhouse until they have reached a height of 12" (30 cm), have a good network of roots and have at least 5-6 leaves.
Prepare the seeds. Stratify your seeds. Plant your seeds. Transplant your seedlings.