Mix a solution of warm water and an oxygen-based bleach, such as OxiClean, Clorox 2 or Tide Stain Release.
Make sure you submerge the garments completely in the solution. Allow the garments to sit overnight.
Wash the clothing as the label instructs but add 1/2 cup of distilled white vinegar to the rinse water. If you prefer, you may add bluing to the wash or rinse water so that the clothes appear brighter. However, you must make sure you follow the instructions on the bottle.
Check the garment to make sure it has properly whitened. If not, you may use a color remover to strip away any color from the clothing. However, avoid using color remover if you have colored trim or decorations on the garment.
Use chlorine bleach on cotton fabrics, if desired. Keep in mind that overusing bleach causes yellowing.
Dry garments outside in the sun if possible so that the ultraviolet rays help brighten the clothing.