Choosing Your Specs
Tank/PvE DPS is likely to be far and away the most popular combination. Those who prefer to DPS may still find situations where they're the only available tank or where they could be useful as an off-tank, while off-tanks may find they want to switch to their DPS spec for trash or bosses where their tanking isn't needed. This is also perfect for a raid DPSer who may need to tank to get together some 5-man runs on off-days.
Tank/PvP DPS is another one that's sure to show up. If you spend most of your raiding and grouping time tanking, you may find you don't really need a PvE DPS tree. Even a tank specced Death Knight can turn out respectable DPS, after all. So that second slot may be useful for PvP DPS. If you like to Battleground or Arena when you aren't tanking, the ability to switch over to to a spec that takes more PvP-oriented talents could be nice and convenient. Plus, your PvP DPS build might even suffice to do some decent PvE DPS damage in a pinch if needed.
PvP DPS/PvE DPS - If you really, absolutely abhor tanking, or if you're a PvE DPSer who enjoys a bit of PvP on the side, you may actually find a double DPS combination the way to go. It's not the way I'd go personally. I enjoy tanking a lot, plus I'm a proponent of being prepared and familiar with as many aspects of your class as possible, so I'll always want that tanking build waiting in the wings. If you're absolutely, positively sure that you'll always have someone to tank though, this may be worth it. Certainly, you should never underestimate the ability of a few talent points (Such as, say Endless Winter or Virulence) to make or break a PvP venture.
Choosing Your Trees
Once you have your roles picked out, it's time to start choosing which trees to use. It seems likely that it will be mostly wise to keep the same tree between your two choices. There's a couple reasons for this. First, there's the raid buffs. If you provide your raid with a certain aura or an AP or Haste buff via talents, people are going to notice if that switches off. If you're taking a dual spec with the intention of changing mid-raid, you'll probably want to take to your raid leadership and make sure your second build still has any group utility your raid will need to succeed.
The second reason is simple familiarity. While the playstyle of a Death Knight doesn't vary too severely from tree to tree, there are specific ability rotations for various specs and trees. Staying in the same tree means you have familiar abilities and can use familiar rotations. There will be tweaks even in the same tree, of course, but it's definitely more convenient not to be switching constantly between multiple similar-but-slightly-different playstyles.
That said, there's also arguments for switching trees between the specs. Take Blood, for example. You can actually get some pretty decent DPS out of a 51/20 Blood build, but when it comes time to tank, Blood is considered strictly 3rd-class in tanking. It has fewer AE threat options, and Vampiric Blood doesn't stack up well against Bone Shield or Unbreakable Armor. Likewise, a tank of any stripe may end up strongly considering a powerful dual wield DPS build of the 32/39 type. Even with repeated nerfs to Killing Machine, it seems to come out on top most of the time, and if you have two solid one-handers, you may just want to dive into being the best you can be.
Of course, if you want to stick with the same tree for both PvE DPS and Tanking, you probably can. Given current content, a solid build from any tree can at least hold their own. That may change in Ulduar, of course, but for now, it's not as much of an issue as it might be. The key is making sure you choose talents that are focused toward your role and finding a rotation that uses your runes and runic power as efficiently as possible.
On the Monkey Wrench that is 3.1
You may have noticed by now that I have talk somewhat in general terms this time around, and avoided linking builds. This is for a good reason, namely, that it's all going to change in 3.1. Even the builds I have linked will probably not look the same and will likely perform differently in 3.1. For example, we know that the Frost Tree will be switched up significantly. Not only should this affect PvP builds as Endless Winter gets moved to the shallow end of the tree, but I expect dual wielders may need to regroup after Killing Machine moves deeper. Likewise, Unholy's Gargoyle/Unholy Blight swap may cause Blood and Frost knights who take 21 in Unholy for the Gargoyle to reexamine their specs. Plus, there's sure to be more changes we don't yet know about.
So in short, while you can make some solid decisions now about which 2 roles you want your talent trees to fill, solid decisions on which trees and talents to take may need to wait until at least the 3.1 PTR. Change is coming, and we're going to need to take it all in before making an ultimate informed choice. Even the all-purpose Death Knights need some specialization, so dual speccing should be as useful for us as it is for anyone if applied right.