To really brighten a white tablecloth, fill a large pot with water to which you've added a sliced-up lemon, peels and all. Bring the water to a boil and then remove the pot from the heat. Add the tablecloth to the pot, let it soak for about an hour, then remove it from the pot and launder it as usual. You might consider hanging the tablecloth outside on a laundry line to dry, as well, since sunlight naturally bleaches white fabrics.
To brighten smaller, white-linen items, such as napkins and handkerchiefs, first wash them as you usually do, then soak them overnight in a sink or tub of water to which you've mixed in a small amount of cream of tartar. Rinse the linens with plain water and, if possible, hang them in the sun to dry (or simply dry them as usual).