Good evening John,
Others have covered the profile question, and I would just add some commentary on pricing and choices.
DRC has always had a lot of brand value, but right now that brand value is so high that once you are willing to spend the money for a particular DRC wine, the relative difference in price between a decent and great vintage is not going to be all that much for the wines that are at or near the $2K spend you are targeting- namely Echezeaux, Grands Echezeaux and Romanee St. Vivant.
$2,000 is a difficult price point. You can get in at that level at retail, but your choices will be very limited and with DRC it is always a good idea to pay something of a premium to get a bottle with impeccable provenance. If you could up that to $2,500, then you will find you have a great many more choices plus you can be picky about provenance. Counterfeiting is always a concern with DRC- but your greater concern is heat damage. As the market shows you- these bottles trade hands at auction quite a bit. All that shipping just leaves room for the possibility of a bottle getting a little warm at some point, and- while not ruined- maybe being slightly impaired. The good news however is that starting with the 2001 vintage, DRC wines have tended to sell through quickly at retail- plus high end retailers by this time tended to have cool rooms for storing top wines, and so your odds of getting a lightly heat damaged bottle due to it sitting out in the 70s for a year or more is far lower than it used to be.
As for the specific choice, again I would suggest being willing to spend a bit more to get a bottle that is likely to show incredibly well. For a first-timer, I think you are best going for a wine that is just entering maturity or on the upward trend within peak. Going younger also means you avoid many of the risks I note above. But not too young because if the wine is shut down you will likely be very disappointed.
And given that Echezeaux, Grands Echezeaux and Romanee St. Vivant are the ones most likely to be in your price range- I think your target vintage range should be 2000-2008.
Be careful about the 2000s though. The Echezeaux is lovely- that one I would absolutely recommend. Romanee St. Vivant when I last had it 10 years ago was very nice, but also very meaty and somewhat foursquare. It lacked nuance at the time. La Tache is beautiful- endless cascades of rose petals- but it is on the light and ethereal side for La Tache. I like that- but many do not.
2001 is the vintage I would go for. The Echezeaux is splendid- a big step up from the excellent 2000 but only $100-200 more expensive in the retail market these days. And Romanee St. Vivant is out of sight good. It is a real knockout and along with the 1993 one of the greatest DRC Romanee St. Vivants I have ever tasted. It was unfurling splendidly a decade ago and should be in full cry now. Even within the range of DRC- the 2001 Vivant has a wow factor that is rare, and if you want to try a DRC for less than $3,000 just once in life- this is the one to try now I think. Even if you had no limit on your budget- this one would be on my list.
The 2006s are superb across the board- but that vintage in both Burgundy and Bordeaux had a long honeymoon phase followed more recently by in many cases shutting down harder than I would have expected. If someone with very recent experience says 2006 is a go- then consider it. But I am not touching those myself right now.
I have not tried the 07s or 08s- 2006 is where my nearly every vintage experience ends and gets spotty- but based on my general knowledge of both vintages, and even moreso since Kevin and others here who know there stuff have recommended them, I would say either vintage is a good choice- but I would go for the 2008 over 2007 unless the 2007 is far less expensive.
Anything else in the vintage range I suggested is either not as good- though just as expensive- or too soon to drink. The 02s might be coming around by now, but after my last experience with Grands Echezeaux I would not bank on it. 03/04- not worth the tariff IMO, and I would not risk opening 05, 09 or 10 this young.
Final thought- I would suggest Echezeaux or Romanee St. Vivant from the 3 that are in your price range. The Grands Echezeaux is a spectacular wine, but it is also one of the longest lived and it can be at times very unforgiving if opened too young. Since you are looking for a one-time experience, I think Ech or RSV will be the safer bet.