The practice of asking the father of the object of one's affection for her hand in marriage is a charming, if somewhat antiquated, practice. (Makes one think of the joke whose punch line is, "Sorry, but you can't have just the hand; you're going to have to take the whole girl!") If the bride-to-be suggests it would be a good idea or courteous then it should definitely be done. If you receive no guidance in this regard then consider that it is now the 21st century and women should not be treated like puppies going from breeder to master, as if they had no say in the matter. Another thing to keep in mind is the question of what happens if Dad says no. Do you break up? Load your shotgun and start a good old-fashioned feud? Not likely, so ultimately it doesn't really matter does it?

Nevertheless, courtesy is not outdated so stammering young suitors anxious to win Dad's favour should seek a parental blessing rather than permission, a course of action that lies somewhere between lodging a formal application and just phoning her parents to announce the deed afterward.
First, confirm with your girlfriend that she does indeed want to marry you by proposing. Then, approach her father, or both her parents would be better, hand in hand. Announce your intent to marry and ask that he/they bless your union and accept you as part of their family. That way, even if dad dislikes the idea, your girlfriend can't be said to defy him if she decides to tie the knot in spite of him.

Asking for her hand, however, puts your fate where it really doesn't belong, e.g. at the whim of your girlfriend's father, who may not have your best interests at heart and may have problems to his little girl marrying anyone, let alone you. Dad doesn't know the intimate details of your relationship (let's hope) and won't have an accurate picture of how good a match you two really are. As well, even the least liberated among us usually likes to be consulted about who she spends the rest of her life with, rather than be handed off from one man to another like a football.
Of course, if you're sure that her answer will be yes -- if you've been together more than a year, things are going splendidly, and you find her lingering in jeweller's windows and net-surfing around weddingworks.com.au, and you know that she's an old-fashioned, traditional kind of girl who would find an appeal to her father sweet and proper, than go for it.

The request is usually made in person, in the bride's parents' home. Call ahead of time and tell her father that you'd like to talk to him soon; don't drop by and run the risk of catching him on his way out to a round of golf. (He could brain you with a nine-iron if he doesn't approve of your relationship.) When you arrive, begin the conversation by explaining your station in life. Play yourself up: your education, your employment, your prospects, your residence. (Yes, this is very corny -- "Here's what I have to offer your little girl," but if you really want to do it the way it's been done...) Say that you would like to share your life with the woman who means the world to you and conclude by asking formally: "Mr. Smith, it would make me the happiest man in the world if you gave your consent for Jane and I to marry."