With that comp the priest is just as much of a damage dealer as the DK. The DK's
https://wotlk.cavernoftime.com/spell=51161 increases priest burst quite significantly. I found in playing this comp that it will always feel counter-intuitive for a priest to start doing damage as, in almost all cases, you're the easier target to beat on. You have to find those windows no matter what. The DK cannot kill a geared healer by himself as there is no healing reduction. Your job is do dispel, line up and burst. All of this has to be somewhat coordinated to work. Main reason why Disc/UH is so much harder to play than Hpala/UH.
As for the DK himself, it's not the most complex job training a target. The hardest thing to do is training awareness and not being trigger happy with your buttons. Here's a few things that I had learned playing this comp. Keep in mind I am not a gladiator DK and have not passed 2.2k on the class so you might look around for more opinions after this one.
1. Always blanket your diseases with
https://wotlk.cavernoftime.com/spell=53640. This is what generates pressure against those pesky dispellers.
2. Use your gargoyle correctly. This means don't spawn it when your target is moving freely and is about to turn a corner or is 100% buffed. Line it up with a trinket or
https://wotlk.cavernoftime.com/spell=53344. If for some reason you swap to
https://wotlk.cavernoftime.com/spell=48266, make sure to spawn it in
https://wotlk.cavernoftime.com/spell=48265 as it inherits the attack speed of
https://wotlk.cavernoftime.com/spell=48265, but not the damage of
https://wotlk.cavernoftime.com/spell=48266.
3. Track enemy cooldowns. You can do this by addons like afflicted or even gladius will do it.
4. Speaking of tracking cooldowns, don't use key abilities such as
https://wotlk.cavernoftime.com/spell=49916,
https://wotlk.cavernoftime.com/spell=47568 and
https://wotlk.cavernoftime.com/spell=47481if you can't force a cooldown or trinket in return.
5. In case this is happening, do not overlook
https://wotlk.cavernoftime.com/spell=51328. We found this to be the only way to at least have a chance against Holy Pala/Warr. It hits like a truck on a fully dispelled target and really catches people off guard. And trust me, after 1700 rating, this comp will be the bane of your existence every 2nd arena game.
6. Unholy DK has 4 interrupts in
https://wotlk.cavernoftime.com/spell=47481,
https://wotlk.cavernoftime.com/spell=47528,
https://wotlk.cavernoftime.com/spell=66017 and
https://wotlk.cavernoftime.com/spell=49916. Use them wisely and casters will weep.
Against a healer team:
The general idea is to start putting pressure on a healer as soon as you can reach it. Apply diseases, 1 death coil to prevent dispels for 10seconds. At this point a trinket will proc so line up for a Gargoyle spawn. Continue with Scourge Strikes/Blood Strikes to get the healer hovering at about 65-75%. At this point the priest should link up with you to dispel the target and start nuking it. If everything lines up, the DK will pop empower rune weapon, strangulate silence, continue scourge strikes/death coil and just before strangulate runs out use Gnaw into Corpse Explosion. This is an ideal case scenario of course, thing will not be so clear-cut so don't be shy to improvise. DK is all about timing and synergy in this comp, takes time to get the "feel" for it.
Against double DPS teams:
Keep the priest alive! Do everything in your power to help you healer survive initial burst as he will be the target nearly all the time. AMZ, grip, silence, pet stun. Everything in the kitchen sink to help a priest live. If you can withstand major cooldowns from the enemy team, just pick the easier target and slowly wither them down. This is where your defensive play will really be tested. Comps like Enh/Ret and Ele/Destro will really make you sweat.
To conclude, this comp has a steep learning curve, but feels very rewarding when you get things going. That's why most Unholy DKs will choose a Holy Paladin as their healer - it is simply a lot easier to achieve similar results. You can beat nearly any double DPS comp, but it will never be easy until you learn to play defensively while applying pressure. Healer comps you can have some good success against. Holy Pala/Warr, at least for us, resulted in a LOT of losses.
DK is also very gear dependent, you need top notch equipment to remain competitive. Also, keep in mind that the biggest disadvantage of a Disc Priest is that it con only be effective while ahead. It is by far the worst arena healer for straight throughput. So if you find yourself spamming direct heals as opposed to dispelling offensively/doing damage - you are now playing from behind.
I hope this helps somewhat. If I missed something, feel free to ask in this thread and I'll try to reply.