If you don't care for the smell of bleach, here are a few natural alternatives.
1. Let your clothes soak in a basin of hot water and lemon slices
for one to two hours, then wash as usual. If clothes are particularly dingy, boil the water, turn off the heat, add your clothing and lemon slices, and soak overnight.
2. White vinegar works well, too. Pour 1/2 cup (125 mL) to 1 cup (250 mL) into the wash with detergent. It will whiten, help wash away detergent or soap residue, and soften the fabric.
3. One more idea: the old-fashioned power of sunlight helps brighten whites and gives them that wonderful outdoor scent.