Summarize:

Look for bright, green leaves without brown spots. You can purchase your basil or pick it from your garden. Remove the leaves from the stems, discarding the stems. This method tends to bring out a stronger flavor in the oil, as the basil is cooked in the oil rather than in the water. In a blender or food processor, pour in 1 cup of oil, and add the bunch of basil leaves. Blend the mixture until you have made it smooth. Place a small skillet on the stove, and turn it to medium to medium-high heat. Pour the basil oil into the skillet, and let it heat for about 3 minutes. Take the pan off the heat. Do not let the oil boil. It should stay at a light simmer, so turn down the heat if the oil is getting too hot. Place a fine mesh strainer over an airtight container with its lid off. Pour the oil through the strainer. Don't push on the pulp, as it can make it fall through the strainer. Gently shake or tap the strainer to help the oil through. Place the oil in the refrigerator. It should be good for about a week.
Choose a fresh bunch of basil. Add the basil and oil to a blender. Heat a small skillet. Add the basil oil. Strain the oil. Keep the oil in the refrigerator.