It is possible to grow grapes from seeds harvested from another grape, however it is unlikely that the resulting plant will replicate the original. If you do have, or know, an existing vine that you like, then taking cuttings is the best way to get another. It is not recommended to attempt to grow grapes from store-bought grapes as you will not know if the type of grape will suit your climate and the resulting plant will not produce grapes of the same quality.