How can you brighten the whites of your eyes?

Hmm... depends. On why you think they're not bright. The sclera (white part of eyeball) can be yellow, due to liver disease. The conjuntiva which covers the sclera can get infected or irritated, and appear red (conjuntivitis). Blue sclerae = bone/connective tissue disorder. Otherwise, an ophthalmologist should assess the eyes but please don't put anything in your eyes to do this on your own!
You can't. There isn't medication to make your eyes look whiter although eye drops temporarily shrink blood vessels in the white part of your eye & make them look whiter. However, your symptoms will worsen if you use the drops on a regular basis. Certain cosmetics can also improve the appearance of your sclera, or the white part of your eye. If the eye is painful, yellow, or draining, see a doctor.