The most efficient way to build up a collection is to buy cheap cards from the bots. They'll sell junk commons for around 128/tix.
However, the vast majority of competitive tournaments/games that you'll be playing in are going to be a limited format, meaning that you only have the cards you open during the game (such as in drafting which you mention). Pre-constructed tournaments are quite rare in Magic Online; and so your collection will not have much input onto your games.
That's not to say that having a large collection and good knowledge of the current sets isn't important in the limited formats (it is!), just that the size of your collection itself doesn't mean much in the competitive arena.