To Activate this Quest, you must complete the quest Bruma Gate from the main story. After speaking with Jauffre, who asks you to gather support from all the other cities of Cyrodill, and so the quest begins.
Anvil
Go to Castle Anvil, speak with Countess Millona Umbranox. Ask for Aid for Bruma, she is willing to help, but not with an Oblivion Gate just outside the city walls. If you have closed the gate already she will lend support.
The Oblivion Gate
Just northwest of Anvil is the Oblivion Gate. Once in the Wastes of Oblivion follow the path southwest. Continue along it southeast, and when it you reach the fork in the road, take the southwest path. Never mind the two towers, keep your eyes on the west wall from this point on. There is a doorway kind of camouflaged into the wall called Nether Tunnels, head on in.
Take either tunnel to be on your way, it doesn't matter which, and head southwest. The door, Oblivion Cave, shouldn't be to hard to find. Inside, follow the north path and head down the northeast tunnel after that, head down this route and enter the Oblivion Cave. This leads back out to the Wastes of Oblivion.
Back outside take the northeast route. At the end of the path is a doorway to The Red Gnash Channels. Inside again, take the northwest trail and follow it until you see opening along the northeast wall. Go in and follow the path to its end, where a doorway leads back to Wastes of Oblivion.
Take either Rending Halls doorway and follow the path to a room. There is an opening along the north wall, enter and follow it until you reach The Flesh Spire doorway. Head up the ramp until coming to a door along the south wall leading to Dead Halls. Go up the path and until you find a small room. North will be a path which will bring you up, take the path, at the top is another room accompanied by a couple doors along the south wall which will bring you to The Flesh Spire.
Finally, follow this spire all the way up until you reach the final room, Sigillum Sanguis. Go up the spiked staircase on either side of this area and then up the red ramp. You are at the top of the tower now, head to the center and remove the Sigil Stone.
Head on back to Castle Anvil and speak with Countess Millona Umbranox, picking Aid for Bruma to complete this part of the quest.
Bravil
Cheydinhal
Chorrol
Go to Castle Chorrol and into the great hall. Talk to Countess Valga and ask about Aid for Bruma. She too is willing to help, but not with an Oblivion Gate just outside the city walls. If you have closed the gate already she will lend support.
The Oblivion Gate
From the gate head down the south path and you'll notice that your path is blocked by a Tsunami Gate. Back track towards the Oblivion Gate and hang a left and look along the east wall for the entrance to the Tsunami tower. Use the Corpse Masher to head up the tower. Head up the tower and find the Tornado Gate Control and hit it, this will open the Tsunami Gate.
From here exit out the bottom of the tower. Go back west when outside, towards the Oblivion Gate and head back south to the Tsunami gate. JUST to the west after the gate is a door to the Hurricane tower, head on in. Use the Corpse Masher to head up the tower, there will be a door about halfway up the tower.
Cross the bridge outside bringing you to the Portal of Natural Disaster. You can finally start the climb to the Sigil Stone. Inside is a door south of yours which will bring you to the Rending Halls. Go straight when you are in the Rending Halls through the Citadel Hall Door and head up the path, being careful of the dropping blades from the ceiling. At the end of the path are two doors both leading back to Portal of Natural Disaster.
Up the ramp in here is a door leading to Corridors of Dark Salvation. Inside take either path all the way up, it will bring you to a room. Along the east wall is another Portals of Natural Disaster doorway. Head up and enter either doorway to the Sigillum Sanguis. Go up the spiked staircase on either side of this area and then up the red ramp. You are at the top of the tower now, head to the center and remove the Sigil Stone.
Imperial City
At the Imperial City Palace, enter the Elder Council Chamber and find High Chancellor Ocato. Ask him for Aid for Bruma. He will deny you troops, completing this part of the quest.
Kvatch
Head to Kvatch and work your way through the ruined city to the Kvatch Castle Great Hall. Here you will find Salvian. Ask him about Aid for Bruma to complete this part of the quest.