I've played a Barbarian since release now, read several forums for Barbarian discussion from time to time, and I'm in Hell difficulty right now.
I will give you a break-down of what you, and potential readers of your question, have to expect in the future when playing a Barbarian.
All the way through normal, Barbarian was a blast to play as DPS. You don't have to invest heavily in Vitality. In fact, I went with Strength and other stats over Vitality wherever I could, simply because I wanted to be badass. And surely, I was. You will crush the skulls of legions of enemies with your fingertips. What this means is: In normal, play whatever you want, it won't matter.
However, throughout Act IV you might want to keep some gear with more Vitality that you find, since Act I in Nightmare will be a big change. For me, I had around 4k life I think, and champion packs killed me in a matter of seconds.
In Nightmare, the first thing you will probably have to do is change your gear out for some gear that has higher Vitality on it. The exact amount that you need will probably vary with your playstyle and build, but you will have a hard time if you have less than 10k HP by the middle of Act II, and about at least 17-18k at the end of Act IV (I personally had 22k, which made playing really comfortable). You can reach much higher life-numbers in Nightmare, if you want to. I think it's probably possible to reach 30k and a bit further, if you purely use Vitality-focused gear and gems, however, it's probably not necessary. With my 22K, I already didn't do an enormous amount of damage, and it can only get less with more HP.
I'm still at the beginning of Hell, but from the few Champion packs that I have seen, I think that what most people on forums write might become true very quickly: Playing a DPS-focused Barbarian in Hell is very hard, and impossible in Inferno. When you get to Inferno, simple "white" mobs will hit you for 30-40k. This means that you have to have a lot of HP to survive for enough time to make a difference. This usually means that you have to favor Vitality over Strength wherever you can, and you also pretty much have to use a Shield + 1H weapon, since the lost survivability with a two-hander or two one-handers is usually not bearable. However, I can not give exact numbers on what HP-numbers you need to reach in the various Inferno acts, since my experience in that difficulty are just theoretical (reading in forums) for now.
Another thing I'd like to mention, which probably is true for all classes, is that you should also look into raising your actual resistances (physical resistance + elements), since they are pretty easy damage reduction, especially early on. I have around 100 or so intelligence on my gear, just because some good gear that I needed happened to have intelligence on it as well, and just from those little points and War Cry + the rune that raises your resistances, my damage from everything is reduced by a little over 10%, which is pretty huge. Since the rune also increases your resistance percentage, instead of a flat value, I'm assuming that it will become much more important later on than, for example, the rune that raises your HP by 10% (remember: Essentially, taking 10% less damage is the same as having 10% more HP).