It is not possible to become fair by drinking milk. The colour of your skin is determined by your genetics and how much time you spend in the sun.
The myth of drinking milk for fair skin might come from stereotypes of the 'Milky Bar Kid' being so blond and pale, and this was then associated with the milky white chocolate. Milk is very good for you in terms of the calcium that it provides, however, as with anything, you should always consume it in moderation. Milk also provides Vitamin A, C, and D, riboflavin, thiamine and amino acids. So, it has many benefits, unless you have a lactose intolerance, in which case steer clear. Either way, I'm afraid milk is not a source of a fair complexion.
According to History, Cleopatra bathed in a bath of warm water with milk added, along with honey, orange and lavender. This is thought to aid the removal of dead skin cells and allow bright shiny new ones to show through. However, there is no evidence to suggest that anyone became fair by drinking milk.