Tribunal:Helseth's Champion

Quick Walkthrough
#Speak with King Helseth in the Royal Palace Throne Room and agree to duel with Karrod. 
#Loiter or leave for 24 hours and come back and talk to Helseth to arrange the fight. 
#Fight with Karrod until he surrenders. 
#Speak with Helseth again to receive the Dagger of Symmachus as your reward. 

Detailed Walkthrough
King Helseth
Return to King Helseth and offer your service to him once again. He will wish for you to prove your worth before he gives you any further tasks by fighting his personal bodyguard Karrod. You are to return to the Throne room tomorrow for the duel. Since Karrod cannot speak you won't get anything further from him.

The Art of War
You have 24 hours to kill before the fight so use the time profitably by, for instance, stocking up on healing potions. It is also worthwhile visiting the Winged Guar tavern in Godsreach and speaking to the bartender. He will offer to send one of his girls to Karrod to tire him out, so that he will be weaker for the fight the next day. Doing this will lower Karrod's Short Blade, Block and Medium Armor by 20, his Strength by 40, his Speed by 30, and his Health and Fatigue by 100. This will also remove his ability to regenerate health.

The Fight
The next morning, return to the throne room and speak with Helseth again to arrange the duel. Karrod has a large amount of health and has an impressive set of combat abilities, but interestingly enough he doesn't hit very hard. Once his health is lowered to around 10% he will surrender and Helseth will stop the fight. If you kill him by any means, Helseth will order the guards to attack you, so refrain from using spells with long-lasting effects as they will continue even after the forced dialogue is initiated.

Speak with Helseth
Speak with Helseth after the fight to receive his incredulous congratulations. Be sure to choose the fight my champion topic again to complete the quest. You receive the Dagger of Symmachus as your reward.
Speak with Helseth again about his plans for you, recent events and learning more about the attacks for further instructions.

Notes
It is possible to kill Karrod in one hit if you are using a powerful weapon with a cast-on-strike damage effect such as Goldbrand or Eltonbrand and are highly skilled in its use. Consider changing weapons, if this is a possibility. However, killing him may not actually kill him as the text box popping up regarding his surrender will bring him back to life instantly. (Similar to how mortally damaging a guard and paying their fine may not kill them after the fine is paid.)
If you kill Karrod without anyone noticing before the duel starts (and you ignored the warning about the death of an essential character), you'll find the Odd Dwemer Weapon on Karrod's body, which is needed to complete the Main Quest. If you speak to Helseth, he will simply call all of his Royal Guards to kill you. 
If you just attack Karrod (outside the duel), a Royal Guard will arrest you stating that the King expected some treacherous actions, and will escort you to prison for one day. You can then talk to Helseth and resume the quest.
If you taunt and kill Helseth at this point, he'll have the Dagger of Symmachus on him, along with the extremely useful Royal Signet Ring. While you will receive the usual message about killing an essential character, you can continue the main quest without penalty, with not many bemoaning the death of the King, but you will also be unable to obtain the final journal entry for this quest, along with the full suit of Royal Guard Armor after you have completed The Mad God.
It is possible to disintegrate his weapon during the duel using the Disintegrate Weapon spell effect and then pickpocket it from his inventory after the duel is over.
Right before the duel all NPCs in the room will move away to the corners of the room. However, unless you are invisible, they may stop to say their usual greeting and won't move on, in which case the duel will not begin, as the game expects the AI to finish the movements before turning Karrod hostile. If you wait long enough, they will continue their AI movement after some time, however, this interruption may be confusing to you at this point, as it is unclear why the duel is not initiated.
Due to a bug, Karrod will take additional damage after the fight is over if you cast a health damaging spell on him that lasts for several seconds.
If you have waited 24 hours in the throne room and King Helseth keeps telling you to come back later, try leaving the throne room and re-entering.  Only when you enter the throne room 24 or more hours after Helseth has arranged the duel will the King let it begin.
The quest actually ends when you come back to King Helseth after you finished the Main Quest (so, after The Mad God quest).

Quest Stages