Talk to Ebony Warrior
After reaching level 80, in one of the cities you should be approach by a peculiar character - the Ebony Warrior. He will tell you that he's been searching for years for a worthy opponent. Only death form the hands of such a powerful warrior like you will let him reach Sovngard. Apparently you will be the who gets the honor of killing him.
Meet the Ebony Warrior on the Last Vigil
Go to the Last Vigil found southeast of Shor's Stone and northeast of Riften (screen above). That's where you will meet with the Ebony Warior. The fight will begin with your first attack, so it would be good to begin with a stealthy one.
The fight isn't easy - your enemy is supremely armored, can deal great damage with his Ebony Sword (which additionally drains life) and knows shouts which can send you flying. Don't try fighting him on distance - his bow is surprisingly powerful, dealing both physical and ice damage. What's even worse, he's also 50% resistant to fire, ice and shock - mages can have a tough time here.
The best tactic is bringing along a very powerful companion (Serana will do just fine) and convincing a Dragon to help you (by using the Bend Will Shout). If the Ebony Warior has to fight three enemies at once, he will become a much easier target. You can also boost your abilities by using the Dragon Aspect Shout. It's also a good idea to make use of the special abilities offered by the Black Books.