Dear me, this one was frustrating. There are a lot of working components to getting this, so I'll list them in the order they need to be done, and include the locations you need to go/find people/find key items.
1) You need to get Ascension McDade into power. Before doing this, visit Heidi's brothel to get her quest regarding Veronica. You'll need to kill a dude named Schwag and obtain his ledger as evidence over in the diner. DO NOT TELL HEIDI VERONICA LOVES THE MASKED MAN. Return to the brothel and speak with Veronica and learn about a masked man she's in love with (this is McDade). Now, you can find McDade in Bastion. You get to this location through the main Hollywood map. Head due north upon entering from the Wasteland. There you will find computer Mathias blabbing at you. Just to the left of that there is a mound of rubble. Destroy this with Brute Force or a grenade and enter the tunnel to Bastion. Once through, you'll encounter McDade in the first room of Bastion/Griffith Park. Examine the mask on the table next to him, then question him about Veronica. Ultimately he'll tell you to speak with her again, expressing his love. Befor that, you need to get McDade into power. Travel a little north from him and go the big room to the right. There you will find Fealty Mayweather. Speak with her to get her quest regarding evidence. Return to the room with the two vendors, go outside, and head immediately right to the fence. Use your lockpicker to obtain the eyeball in the case (you can get it even on this side of the fence). Return said eyeball to Fealty Mayweather. She will give you evidence. Give this evidence to Ascension McDade, and there will be a cutscene where he takes over Bastion.
*** EDIT. If you are having trouble with Fealty giving you the quest, Casyle writes that: [Fealty Mayweather never gave me any quests, just a log update to my current quest that I might need to find some way to convince her to help me. All I had to do was get Simm's eye from the display case and then go back and talk to her, then she gave me the evidence.] ***
2) Return to Veronica, tell her about McDade. DO NOT SPEAK WITH HEIDI AT ANY POINT. She will ask you to get her into power. This is fairly easy. You need to convince three key individuals to vouch for her. You will target Flo, Manny, and George Lin. For Flo, simply return to Schwag's diner and speak with her. She'll be glad you killed Schwag and join up immediately. Done. Across from Manny's casino you'll find George Lin and his wife. Simply enter their shop, get the dog to leave, and speak with George and his wife again. With their shop cleared up, George will also join your cause. Your last target is Manny. What you need to do now is go to the park to the east of Manny's casino. It's snuggled there between the casino and the gunshop. In the lower lefthand corner of the park is a well-hidden safe. Use your various skills to open it, retrieve the photos, and give them to Manny. This will blackmail him into joining you. Now that you have three votes supporting Veronica, return to her. This will formally oust Heidi from office, installing Veronica.
3) You will now be tasked with hunting Heidi down in the sewers of Hollywood. Find and kill her and her buddy. While you're down there, be sure to wipe out every last Salt Thug. I don't now if it's completely necessary, but I did it just to be safe. Now exit the sewers and find Ascension McDade outside the entrance of Hotel California. He'll task you with wiping out Retribution Jones and his church followers. Head back to Bastion and go to the chapel area. You'll have to fight the vendors first. Then kill Retribution Jones and his followers in the chapel. Then head outside. You'll have to kill all enemies in this area, and free the penitents on the righthand side. While you're in this area, head up to the radio tower, kill the dudes there, and install the receiver. Return to the main Hollywood area.
4) This last bit took me forever to figure out. What you need to do is head to a different sewer location accessible from the main Hollywood map. You're looking for Swifty's spot. It's down the lower eastern alley of Hollywood, through a manhole. Once you're down there, kill Swifty and his buddies, and free all the slaves. You should get your final log update, informing you to return to Bastion to witness the ceremony. Do so to unlock this arduous achievement.