NOTE** Just a warning, many people I know have waited to do this quest to have an easier time with the monsters in Oblivion. While this may seem like a good idea, the creatures in the game are leveled meaning they get harder as you level up. You can either start this at a low level and fight mostly scamps or you can wait until you're more experienced and fight many Xivilai and Atronachs. It's up to you. I will be refering to any fights with creatures in Oblivion as just Daedra since the types you fight depend on your level.
Following the "Find the Heir quest" of the main storyline, you will be told to find Brother Martin, the last remaining survivor to the throne of the emporer. Speaking with Jauffre at Weynon Priory will give you the information that Martin is in the city of Kvatch. Fast travel there and make your way to the top of the city. At the top, you'll see guard Savlian Matius fighting Daedra that are emerging from the gate in front of you. After defeating the last of the Daedra here, talk to Matius and tell him you wish to help close the gate. He will allow you to enter by walking up to the gate and activating it.
Now you should be in Oblivion for the first time. To your left you should see a survivor fending off some Daedra. Help him finish these two off and he will tell you he is Ilend Vonius and he was part of a group of soldiers sent in to close the gate. Most of the rest have died but one has been taken prisoner. Here you have the option of telling Ilend to follow you or to leave, I would recommend following since it can't hurt to have some help.
Now the two of you are off to close the gate (although in my experience Ilend never survives too long). Follow your marker to The Blood Feast. This will be the first big tower with a yellow light at the top. Make your way towards the tower killing any Daedra that attack you and looting their corpses for anything that could be useful. Enterting the tower, you will see a few Daedra in the circular room. Kill them and feel free to use the Magicka and Health fountains that can be found there. Now enter the Rendering Halls and continue to make your way upwards. At the top of this section you will find the Corridors of Dark Salvation, your next destination. Make your way out of the only unlocked door and you should be on a bridge very high in the air between the two towers if I remember rightly. Enter the tower (Reaper's Sprawl) and you will find the prisoner from Ilend's company and his guard, a Dremora Sigil Keeper. Kill him and make sure to take the key off of his corpse. Now talk to the prisoner for more information about the quest and make your way back to the previous tower.
Now you can unlock the previously locked door and make your way towards a portal which should take you up to the top balcony area. From here you can enter the Sigillum Sanguis where you can close the gate. Make your way up to the top of the wing shaped ramp to find the glowing sigil stone. Sigil stones are the key to closing each Oblivion gate and each one has special powers that can be used on most weapons and apparel. If you wish, save before removing the stone until you get one whose powers you like. The achievement unlocks upon removal of the stone.