You get the quest Alduin's Bane after completing the quest "Elder Knowledge" and finding the Elder Scroll.

In Elder Knowledge you met the master of the Greybeards, the dragon Paarthurnax. He told you that the ancient Nords used an Elder Scroll to cast Alduin forward in time. Now that you've recovered that Elder Scroll, you may be able to learn the "Dragonrend" Shout directly from the ancient heroes.

Open your quest journal:

Alduin's Bane
I've located the Elder Scroll that was used to cast Alduin forward into our time by the Nord heroes of old. If Paarthurnax is right, if I read it near the Time Wound, I will get a vision of the past, when the ancient Nords used the Dragonrend Shout against Alduin

Read the Elder Scroll at the Time Wound

Fast travel to the Throat of the World. As you approach, Paarthurnax will sense that you have the Elder Scroll. He tells you to go near the time wound and fulfill your destiny. But do not delay, as Alduin will be coming.

Locate the objective marker, which is pointing to the time wound and stand there. Bring up your inventory and read the Elder Scroll.

Your objective now is to learn the shout "Dragonrend" from the Nord Heroes

Experience a vision of the past, or perhaps you are in the past. Watch how the events leading to Alduin's demise unfolded. You can control where you look as you watch the historic moment.

A dragon will say that today Alduin's Lordship will be restored. The Nord heroes slay that dragon.. Gormlaith and Hakon now expect Alduin to come. They are prepared. They have a plan. Hakon is worried that Alduin will not come, but Felldir assures him that Alduin will show up. Gormlaith is confident of their ability to slay Alduin, since they have Dragonrend now. Felldir will say that Alduin cannot be slayed like a lesser dragon, he is beyond their strength. Felldir now reveals that he brought the Elder Scroll to even the odds. Hakon will protest, they agreed never to use it.

Suddenly Alduin approaches. The heroes will speak the Dragonrend shout, "Joor Zah Frul" and you will learn it!

Word of Power: Mortal, Dragonrend
Word of Power: Finite, Dragonrend
Word of Power: Temporary, Dragonrend

Alduin drops to the ground. But he is still fighting strong, and tosses Gormlaith on the ground. Hakon urges Felldir to use the Elder Scroll. As Hakon is fighting and distracting Alduin, Felldir reads from the Elder Scroll. Felldir banishes Alduin who disappears in a ball of energy.

You are back in the present now. Check your quest log:
"I read the Elder Scroll near the Time-Wound, and learned the Dragonrend Shout in a vision of the Nord heroes fighting Alduin in the distant past. Now I need to use Dragonrend to defeat Alduin in the present time.

Objective: Defeat Alduin

Alduin has sensed what you are doing and comes to stop you, but it is too late as you have already learned Dragonrend. Equip the shout and prepare to use it against him.

Paarthurnax will help you against Alduin. But Alduin is invulnerable except while under the effect of Dragonrend. Aim the shout and use it on Alduin. When he is high above, you wont be able to hit him with Dragonrend, so wait until he lands or is flying low. When you hit him with Dragonrend he will become weak and fall to the ground,  have at him with all you have got. Make sure to keep using Dragonrend again and again as soon as it recharges, to keep Alduin from regaining strength.

As you are about to defeat him, he proclaims that he cannot be slain here, by you or anyone else, as he is Alduin, Firstborn of Akatosh. He then rises and flies up to the sky and away.

This completes the quest "Alduin's Bane" and start the next quest in the main campaign, "The Fallen"