Is it worth the extra cost and time it'll take to complete Dragon Age Origins? Is there that much back-story I'll miss out on in Dragon Age 2?
From what I've heard, there isn't much impact on Dragon Age 2 from DAO. The story takes place in a different part of the world, and there are only a few side characters that carry over at all, none of which are in your party. So from just the standpoint of enjoying DA2, I'd say no.
On the other hand, DAO is a great game, and DA2 has been getting pretty mediocre reviews, so one could make the argument that you should play DAO first just because it is a better game for less money. =)
While it's true much of what you do in Dragon Age II isn't directly connected to the plot of Dragon Age: Origins, the game assumes you are familiar with the setting established in Dragon Age: Origins so much so that it completely glosses over a ton of exposition which serves to give meaning to the motivations of the main character and the overall story.
That is, if you don't play Dragon Age: Origins, you're going to find yourself asking "Who cares?" more often than not while the plot unfolds. Dragon Age II will give you the CliffsNotes version of a plot element, but the background behind that plot element is fully explored in Dragon Age: Origins.
Additionally, there's a ton of dialogue points in sidequests and with companions which make reference to the events of Dragon Age: Origins that'll be lost on the person who doesn't play it.
But perhaps the most important reason to play Dragon Age: Origins before Dragon Age II is that it's the better game. Dragon Age II has better gameplay and a much improved user interface, but it's not the Godfather II of the series. You'd be doing yourself a disservice by not playing Dragon Age: Origins in order to play Dragon Age II exclusively.
At the time this question was asked, the chosen answer was a good one. Now, however, with Dragon Age: Inquisition having been thrown into the mix, I think it deserves revisitation. It's true that the events of Dragon Age II are not directly tied to those of Dragon Age: Origins. However, if you really want to understand what's going on in DA:I, you have to know the stories of both.
DA:O sets up the grand lore of the setting, focusing on blights, dragons as elder gods, the Grey Wardens, the Fade, magic, and the politics of each of the major races, giving glimpses into their societies. The DA:O expansion, Awakening, gets more into the spirits of the Fade and some dark stuff that's happened in the history of the Grey Wardens. DAII is a much more localized story than DA:O, focusing mainly on the tensions that can arise between mages, who are segregated from the general population in many places, and the templars who stand between them and everyone else.
It's easy to think that all of these things are separate and playable alone, but DA:I weaves all three of them - because Awakening turns out to be important to DAII and DA:I later on - into one coherent narrative. Technically, DA:I can be played alone, but you're doing yourself a disservice if you go that route. The OMGWUT moments only pop if you've played everything.
In terms of gameplay, DA:O really shows its age these days. It's a shame, because it really is a great game with a great deal of interesting lore to offer. DAII is deeply flawed, having an extraordinary amount of reused environmental content. The combat gameplay is different enough that it was quite controversial at the time it came out, too.
TL;DR: Play all the things. Or at least watch a let's play to get the story.
When starting a new game in DA2 besides importing a save you will be able to choose one from some possible outcomes of DAO. Generally not playing DAO does not give you a chance to kill some characters and prevent them from appearing in optional quests in DA2. Besides that the plot in DA2 is independent as it concerns the main character and locations so the events in DAO do not contribute much to the development of DA2.
It's definitely worth playing to get the background, as others have said, but if you're largely concerned about the saved file import you could use the Dragon Age Save Game Creator to generate a Warden to your liking. (Spoilers for DA:O, obviously, as the creator asks you for your decisions at key plot points.)