Destroying the Great Gate outside of Bruma is very similar to closing any other Oblivion Gate except the region of Oblivion that you enter is somewhat larger and you’re facing a fifteen minute time limit, which begins once you enter the gate. Upon entering the gate, you’ll see the siege engine in front of you, the main tower in the distance, and six smaller towers, with three on either side. You’ll need to enter one of the two smaller towers that’s furthest away from you to trip the switch that opens one of the wargates and gives you access to the main tower.
Start by running west along the lava shore until you reach the Smoke and Scorch cave system. This system will allow you to access the two smaller towers. It doesn’t matter which one you enter; simply fight your way to the top and flip the switch to open the wargate. Exiting out of the door halfway up the tower will take you out onto a bridge facing the main tower. At this point, simply fight your way up to the top of the tower. Given the constraints of the time limit, its extensive fighting is not recommended – simply run past as many enemies as you can. Once you grab the Great Sigil Stone, you’ll be transported back to the battlefield outside Bruma and the achievement will unlock.