Speak to Serana. Ask her what you should do now. After speaking with her mother in the Soul Cairn, Serana is not sure that Lord Harkon can be trusted.

You can suggest the following options:
Then we kill him, before he kills us.
We have the bow, he'll listen to us
We should honor our agreement with him.

If you suggest trying to reason with Harkon or honor the agreement, she refuses. Harkon will betray you the second your back is turned. He has to die. We have no choice. (Whatever you tell Serana, you will still end up having to kill Harkon)

Tell her that you will face Lord Harkon, together.

The objective updates:

Kindred Judgment
With Auriel's Bow in my possession, it's time to confront Harkon and reshape the prophecy as I see it.

Confront Harkon with Auriel's Bow

Before heading out to the final fight with Harkon, you can ask Serana about the vampire blood and the bow. She tells you that it is the last bit of the prophecy. The blood of a Daughter of Coldharbour can corrupt the bow. She gives you permission to use her blood, if you have arrows of elven quality. Tell her you would like to treat some arrows with her blood. Only Elven Arrows can be used for this. This will produce:

Bloodcursed Elven Arrow
Shrouds the sun in darkness if shot with Auriel's Bow.

Also, make sure you have plenty of Sunhallowed Arrows.

Take the portal to Darkfall Cave (From where behind where Auriel's Bow was). There is a tunnel to the Southwest leading to a pull chain. Activate the Pull Chain to reveal a secret passage leading you close to where you entered Darkfall Cave earlier. Go west, then north, activate another pull chain, go past a chamber with a waterfall, and exit the cave back to Tamriel. Fast Travel to Castle Volkihar.

Enter Volkihar Keep. Make sure you have Auriel's Bow. Head to the Volkihar Cathedral.

Skyrim Dawnguard Harkon

Listen to the conversation between Harkon and Serana. Harkon then turns to you. When the first dialog wheel comes up, your choices will be:

I would never use Serana like that
No, I'm here to prevent our annihilation
The bow belongs to me now

Try he third option. Lord Harkon appreciates your honesty and asks you how it feels to hold the fate of the vampire in your grasp. He demands you give him the bow.

Your choices are:

Very well. (Give Auriel's Bow)

Choose the first option. Your objective now is to slay Harkon.

What happens if you give the bow to Harkon?
If you foolishly agree to surrender Auriel's Bow to Harkon, he betrays you and will still attack you. Only now you would be at a disadvantage, since the bow will be use AGAINST you!

Lord Harkon Strategy - Final Boss Fight
Harkon summons Skeletons and Gargoyles (inclduing Gargoyle Brutes) to help him. Let Serana handle the adds, and you focus only on Lord Harkon
Harkon casts drain spells on you, as well as transforming into a swarm of bats and teleporting from one place to another.

Skyrim Dawnguard Harkon Blood Shield

After being hit a few times, Harkon teleports to the Altar and casts some sort of Blood Shield. Only with Auriel's Bow equipped, armed with Sunhallowed Arrows, can you penetrate that shield.

Use Auriel's Bow to kill Lord Harkon in Skyrim Dawnguard

When Harkon is dead, the quest "Kindred Judgment" is completed. Search Harkon's Ashes. Among the loot, you will find Harkon's Sword, Potent Poison, Potion of Blood, Vampire Dust, Black Soul Gem (Grand), Vampire Royal Armor, Vampire Boots, Gold and Emerald Circlet, Circlet of Eminent Restoration, a Necklace of Magic Suppression and a Ring of Eminent Destruction.

Harkon's Sword is a powerful sword that absorbs Health, Magicka, and Stamina if wielded by a vampire.

Skyrim Harkon's Sword

Talk to Serana, she tells you that if you have any more adventures planned, she would love to come along. You can agree and have her join you as your companion.

If you talk to Garan Marethi, he refers to you as the new master. The Castle is yours. The vampires will follow you and help spread your influence across Skyrim.

You can use Auriels bow, armed with the Bloodcursed Elven Arrows, to turn off the sun. This is a temporary effect that lasts a day, but you can shoot the sun again every day to keep Skyrim dark. The effect is pretty cool, so you should go ahead and try it.

Congratulations! This completes the Vampire Lord Storyline in the Skyrim Dawnguard DLC!