I've been playing WoW for nearly 9 years, WotLK for kinda 7, all with a DK which is my favourite class. I've spent time in tons of server, PvE and PvP oriented and I consider myself well skilled with dueling as a DK and if I should mention the class which gives more troubles to DKs i would say "Paladin".
On 1vs1, using a Unholy DK at quite high skillcap, a Retribution Paladin (same skillcap) overpowers you. Of course, if we suppose he's gonna use Divine Shield. If not, he will give a lot of problems anyway. Lichborne? Insta-Turn Evil, then heals. Lichborne + Antimagic-Shell? Kite, preventing RP generation from attacks and heals on proc by Art of War. Gargoyle? Insta-Turn Evil again. If he doesn't bubble-up, at same skillcap, the DK wins with good CD management, but most of people use Divine Shield constantly on duels, waiting for Forbeareance to fade-off, then Avenging Wrath.
On Arena side, a Unholy DK can't kill a Holy Paladin alone, neither using Corpse Explosion on low health, at high ratings (supposing, so, high skillcap) while a Paladin can due to Avenging Wrath which bypasses 50% of ALL damage reduction effects and good crit procs. On the other hand, UH DKs have more utility CDs and a perma-snare with Chains of Ice, that's the reason Holy Paladin and Unholy DK is one of the best 2s comps on WotLK (Elemental Shaman and Protection Paladin too, but it works way better with ToC+ Gear, and other classic comps).
On 3vs3 and 5vs5 a Unholy DK with Wandering Plague can give serious issues to the whole enemy team, thing a Paladin can't do.
Frost DKs are a good choice in Arena only with good arp rating (usually it's used also a 2h weapon for best damage, because the sustained dps of the 1handed Frost DK can't be used at his best), also, Frost DK can't win 1on1 a Unholy DK at same skillcap, due to improved Antimagic-Shell, stronger Lichborne+Death Coil Macro and Gargoyle, mind-freezable only each 10 secs.
So my PERSONAL verdict is the following:
-1on1 (without Divine Shield) = UH DK wins
-1on1 (with Divine Shield) = UH DK loses
-2on2 = depends on team composition and other team member's skill, cooperation, management of CDs
-3on3 = same as above
-5on5 = UH DK can be painful
In addiction, always assuming we're talking of high skillcap, the DK can have problems against Rogues (serious problems if Frost DK), and equivalent higher issues against an Arms Warrior (in both cases, if possible, take Rune of Swordshattering but take care because some Wars tend to have a Glyph of Overpower spec for duels, instead of the Glyph of Bladestorm one).
Also the Enhancement Shaman can be tough: it's all on good CD management, 'cause Enha Shams starts always with their massive burst, during that it's hard to kill them in melee. If the DK uses well Icebound Fortitude, Shell and Anti-magic Zone, with ghoul's stun, he wins, else the shaman does.
Shadow Priest and Warlocks are anti-classed by a common DK, if Unholy there isn't much they can do. Moonkins, Mages and Elemental Shamans (in this order) are a bit harder to kill, but Unholy DK is a beast against casters, especially with Rune of Spellshattering.
Note: duels against Blood-Tanknig Death Knight are a waste of time where they usually win also with a less lower skillcap, Blood DpS Death Knights, instead, lose way more frequently due to the lack of movement.
If I, again, personally, should do a list of difficulty from 1 (lowest) to 10 (highest), mine would be the following [assuming the always mentioned Unholy DK (tbh, I love Frost, but it's a PvP topic)]:
Shadow Priest = 1-2; Destruction Warlock = 1-2; Affliction/Demonoloy Warlock = 2-3-4; Arcane/Fire Mage = 3-4; Frost Mage = 4-5; Fury Warrior = 3-4; Protection Warrior = 6-7; Arms Warrior = 7-8-9; Balance Druid = 3-4; Feral Druid = 6-7; Elemental Shaman = 3-4-5; Enhancement Shaman = 8-9; Protection Paladin = 6-7-8; Retribution Paladin = 8-9-10; Combat Rogue = 5; Assassination Rogue = 5-6-7; Subtlety Rogue = 6-7; Tanking Blood Death Knight = 7-8; Damage Dealer Blood Death Knight = 3-4; Frost Death Knight = 3-4-5; Unholy Death Knight = 8-9; Beast Mastery Hunter = 5-6; Marksmanship = 6-7-8.
I would like to repeat myself: this is my personal list, based on playstyle, experience and such, but with the experience growing, you tend to recognize some things with more objectivity.