Sometimes a dirty neck isn't really a dirty neck. You may have sun-damaged skin, or wrinkles in which oil and dead skin cells collect. You may be over-washing your neck, stripping it of natural oils and contributing to dry, dead surface skin. When you exfoliate dead skin cells on your face, you may forget to exfoliate your neck as well. You may also neglect to moisturize your neck and apply sunscreen as you do on your face.
Exfoliate the skin on your neck when you take a shower to remove dead dry skin cells and grime. Use a commercial exfoliant cream or a handful of salt or sugar with a drop or two of liquid soap to scrub the back of your neck, as well as other areas of skin such as your arms and legs that are exposed to sunlight, grime, oil and sweat.
Bleach your skin gently with hydrogen peroxide or lemon juice applied with a cotton ball. You won't see immediate results, but over a period of time your skin tone will even out and look cleaner.
Moisturize your skin after every shower. Exfoliating not only removes dry dead skin cells, it can also remove some of the natural protective oil from your skin. Replace that oil with a moisturizing cream or lotion applied while your skin is still damp. This retains moisture and also provides a barrier to grime.
Apply sunscreen to your skin every day, including your neck and any other areas that may be exposed to the sun, such as your face, ears, scalp, chest, arms and legs.