(I play this on my pally, so this point of view is pretty biased to that, rather than the DK)
The pressure is really on your Pally, not you. He needs to be aware of any cc landing on either of you to cleanse/sacrifice/divine sacrifice it, he needs to be able to recognise when he can play offensively, when he should play defensively and when he should heal, needs to be able to use the correct heal in the correct circumstance, plea when there is less pressure, needs to move sacred shield around depending on the other teams target choice, and, since he is the one with the cc, needs to call most of the plays as well.
The basic kill/opener is repentance -> hammer -> strangulate on a healer while dps'ing the dps, but there are many situations in which this won't work, and, of course, you have to wait at least a minute in between these attempts (pally should call the strang most of the time, it's wasted if it won't guarantee the kil). This is most likely to work on a shaman healer, less likely to work on a paladin (especially if you catch them in the middle of a heal with a little momentum, or with no ES/SS on your dps target), can work on a priest if you start the combo just before the pennance comes off cooldown and will almost never work on a druid whose dps is pre-hotted.
General rules:
Druid healers - Catch them in a cyclone (can't be repentanced in tree form) to gain enough momentum that they actually have to trinket something and/or switch to a non-hotted target when the druid has no trinket (you can kill the druid in a hoj-strang if there are no hots on him, but his dps will probably have enough cooldowns to survive it unless you catch them off guard). Druids will wreck you if you can't control the game.
Priest healers - Easier to cc, but bring enormous offence to the table, especially given how much ret survivability drops when ss/art of war is dispelled. If you can last to the second hoj/repentance (and they're not playing with a dispeller) you should be able to score a kill there. Going for a priest is generally a waste of time unless you can catch them with no PW:S and no trinket/PS ready, since abolish/no ms completely wrecks your chance of killing them. The real problem with a priest healer is often your Ret going oom due to no mana refunds on shielded targets. The only real advantage you have playing against a priest is that they actually go oom, unlike pallys and druids.
Shaman healers - You have to kill grounding totems for your ret, and cleansing totems for yourself, otherwise you will do very little damage. I often find shamans ES themselves, leading to an enormous amount of damage on their DPS in just the repentance, leading to a repentance trinket, leading to a very easy kill in the hoj-strang on either of them. If you can't pull this off, however, you could be in for a rough time with purgeing of devensive buffs and shaman assist dps. Shamans CAN be oomed, if you have enough luck/pressure, and killing a mana tide goes a long way towards this.
Paladin healers - With no MS, pally healer cooldowns + their dps's cooldowns make it very hard to kill either target. You're probably best off trying for a pally kill, but if their dps is not a dk/doesn't have ss on them, you can blow quite a few defensive cooldowns just in your opener. Remember that focus death grip is an awesome extra interrupt on a pally, especially if he thinks he's safe when the rest of your cooldowns are used and goes for a HL.
Common matches:
Warrior/Druid - If the druid plays well and never leaves tree-form when you have reprentance available, you won't win. Eventually ret healing will fall behind and one of you will die. If you dodge bladestorms well (divine sac/sac/hand of prot/run in opposite directions) it will take that much longer to die, however. Look for an opportunity to kill (usually involves a zero hot switch, which a good druid won't let happen, and a good warrior can peel from). Even a few hots will deny a kill opportunity, and mindlessly training one target the whole game will never work. With no MS, splitting dps is completely futile.
Healer/DK - Generally speaking, DKs should have enough cooldowns (+CoI spam) to survive any cc attempts on their healer, and enough peels to deny some kill attempts. DK teams that get momentum at the start can really wreck you with a strang on your pally + cooldown pops.
Healer/Hunter - Hunters have enough survivability cooldowns to outlast cc (entrapment/frost trap/disengage/deterrence/scatter shot/pin/roar of sacrifice etc) and enough dps to take a third to a half of either of your health every single time they exit melee range (IE every disengage). That couped with mortal strike, cc on your ret and added offence from their healer makes this a next to impossible comp to beat. Make sure your ret knows that until 3.3, he can judge/hammer of wrath through deterrence though.
Priest/Mage - Should be completely unwinnable. Any Deep Freeze, Fear or Poly is a kill opportunity, and you will both spend time out of range on either target, and both targets have multiple cooldowns to keep them alive.
Priest/Rogue - Have your pally kite the rogue to some LoS to avoid priest assist/dispels on the pally. Grip the rogue off whenever your pally gets in danger so he can top himself up. Can go both ways, mostly depending on the opener, but definitely winnable. Teams that go for the pally are the most dangerous, but teams that go for the DK can often outlast you, if the pally is forced to heal too much.
Healer/Warlock - DK on warlock, ret on pet, drag the pet out of LoS and kill it twice. Depends very much on the warlock's spec. If he is destruction, and you can get most immolates dispelled, they will do very little damage to you. That said, if he is destruction and lands a fear/immo/chaos/conflag combo on either of you, you've just about lost the game at that point. If he's affliction, the pressure on both of you will likely eventually result in a kill on either of you
Mage/Rogue - Rush the mage so you both get in combat, if you get sapped, call for a HoP, you really cannot afford to lose momentum against this comp. The kill target is the rogue, and your aim is to prevent a vanish reset. If they get a clean reset off, the mage can easily survive the two of you with kiting/ice block until the rogue is safe to reopen (IE pally cooldowns used/divine shield fades). Can go either way, mostly depending on the opener
Caster/Mage - Blow cooldowns like crazy, kill the target that isn't the mage, abuse LoS if one of you gets CC'd and cannot break it. Immolate/Flame shock dispels crucial.
General 2x dps tips - Don't be afraid to HoP a sap, don't be afraid to lichorne to avoid saps, sacrifice early and run in quickly so your pally can avoid saps, make sure your pally switches sacred shield around correctly depending on the target
A few other tips:
- My DK spends alot of time Death Striking in between Art procs to keep himself up to our next burst attempt, it is a very powerful ability against most teams.
- Communication is completely key. If you split dps, neither of you will kill anything, you need to always be working together. This is especially true when it comes to cc's/cooldowns, your ret needs to know when you need to be cleansed, and you need to know when he isn't in a position to keep you alive, when he needs a peel or when he needs to plea
- Your ret needs HoP/HoS/SS/Flash of light/cleanse macros to target you with
- Momentum can win or lose a game, even against teams you've fought before and beaten/lost to
Final thoughts: While I may sound very negative, this is actually quite a fun comp to play as a ret pally. Luckily, not everyone is a fantastic player, and plays in a perfect manner to counter it

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The World of Warcraft Armory is my paladin, we're at the unimpressive rating of 1810 at time of posting. It's worth noting that we've jumped ~150 rating in the last two-three weeks after my DK finally respec'd UH from Frost.