This quest is first initiated when you use the Call Dragon shout on the Dragonsreach Porch to call Odahviing and trap him. Once you have him trapped and have a conversation with him, you end up getting him to reveal the location of Alduin's portal to Sovngarde and then flies you to Skuldafn.
Once you get to Skuldafn, there will be a number of high level Draugr (wraiths and deathlords) before you get to the door to head inside the Skuldafn Temple. I did not count exactly, but there are at least 8 Draugr (some deathlords, some wights, etc.) and there are two dragons. Be careful not to attract the attention of everyone all at once or it will be a very difficult fight.
After going through the left gate, you will go up some stairs to the left and end up on the walkway above a room with a single Draugr Wight below you. As you approach him, a Deathlord and another Wight will also begin to attack you. After you kill the three of them, head straight through where the Draugr Deathlord came out of. There will be frostbite spiders you need to kill in this area, but they shouldn't be much of a problem. Kill them and continue on along the corridor until you reach the double iron doors. Through these doors are another two Draugr - a Deathlord and a Wight. As soon as you open the doors you will notice another pillar. After killing all enemies you will probably notice that there is no exit from this room. The pillars are what provide the exit to you.
When you go through these doors you will enter a room that has another three Draugr in it. There is a Deathlord, a Wight, and a Scourge. Defeat them and continue up the stairs, across the bridge, and through the corridor to the spiral staircase. At the top of the spiral stairs is a room with three more Draugr and a gate that needs to be opened by lever. directly across the room from the gate is an iron door behind which is flammable oil on the floor which can help kill the Draugr. If you go through that door and turn left 180 degrees you'll see a lever that will lift the gate so you can continue forward.
As you continue forward through the passageway ahead, be careful of the flammable oil trap. When you turn left at the end of the corridor with this trap, straight ahead will be a puzzle door and a Draugr Death Overlord guarding it. Kill the guard and take the Diamond Claw from him to get through the puzzle door. The solution from top to bottom is wolf - moth - dragon (as shown at the right).
This door will allow you to enter a room with a word wall in it. This word wall gives you a word for the Storm Call shout. There are no more Draugr in this section of Skuldafn so you may go straight up to the word wall and then continue on without worries. If you continue toward the quest marker, you will get to a door that says Skuldafn. Be prepared before leaving through this door. After you leave the door, you will be in an area with three Draugr Deathlords, a Draugr Scourge, two Dragons, and a Dragon Preist. Be especially careful of the Dragon Priest as he has a staff that will do 50 shock damage per second. You can get to him before he grabs the staff, however, if you move quickly. He will start heading for it as soon as you go out the door - he is out the door and to the left and then up the stairs. Get to him as quickly as possible, but still try to avoid having all enemies attacking you at once.
Once you have defeated all the enemies, you will be able to get to Alduin's portal to Sovngarde. To use the portal, you need to activate the Dragon Seal, then you can simply walk into the portal. If the Dragon Priest has not yet reached his staff, then all you need to do is to walk into the portal.
Once you have gone through the portal, you will need to make your way through the mist to get to the Hall of Valor. You will meet a soldier quite quickly who will tell you how to get there and will follow you. You need to simply keep walking through the fog and get to the bridge. When you get to the bridge to the Hall of Valor, you will meet Tsun with whom you need to talk and later fight in order to gain admittance to the Hall of Valor.
Once you have killed Alduin, there are some awesome animations as he dies and then you just need to go talk to Tsun again to be sent back to Nirn. When you talk to him and tell him you want to go back home, he teaches you a shout to bring a hero from Sovngarde to aide you - the Call of Valor shout. This is officially the end of the main quest. Congratulations on completing an Elder Scrolls game.