State of Decay

The achievement is called 'Last Voice of Danforth', and it's for the other ending of the DLC.

SPOILERS AHEAD!

Once you have completed Sasquatch's missions, and armed the nuke, continue playing but not evacing until Vienna Cho contacts you and asks you to come rescue her. By that time, I had finished nine sieges and rescued all but one civilian. Pick her up and bring her alive to your base. I strongly recommend making a backup on a separate storage unit to avoid having to replay the entire DLC for the other ending (in case you don't alread have it), and having a fail safe in case Vienna Cho dies.

Talk to Vienna, and she will suggest disarming the nuke. You must drive her to the tunnel, and protect her for a couple of minutes until a cutscene comes up telling her she succeeded. This can be tough, so take your best soldier, backup, and stock up fully.

Once done, return to base and call in evac from the base menu. Survive one more siege and achievement unlocked!

EDIT: It would appear that you need to have all missions done before you get permission to evac. So if you have any open missions, complete them until HQ says you can pull out. Thanks to MMDI for pointing that out.

*** SPOILER WARNING ***
This solution spells out pretty much the whole game.

This secret achievement is called "The Last Voice of Danforth." The description for it is "Escaped Danforth with Vienna Cho in your community." To unlock this achivement, you must save refugees in distress and evacuate them as you perform the story-related high value asset (HVA) evacuation missions. After saving several refugees, Vienna Cho may ask for you to rescue her. Do it, take her to your base, and call for an extraction via the radio. Keep her alive during the final siege, and you will unlock the achievement.

As you know if you have played, Vienna Cho is slow to trust the army. If you do a flawless job in rescuing refugees, her confidence in you will grow, and she will ask for your help sooner. If you lose any, she is more liable to wait. The longer she waits to ask you to rescue her, the more sieges you have to survive. (It is said that even if you lose a lot of refugees but survive enough sieges, she will ask you to resuce her anyway. I cannot personally confirm this yet, but it sounds logical.)

Besides losing refugees, one other thing that can foul this up is Sasquatch's nuke. If you complete all of Sasquatch's missions and then go look for him in the tunnel, his nuke will be armed on a dead man's switch. Vienna won't leave Danforth if that happens. You then have to talk to her and take her to the tunnel disarm the nuke. Other than this, it doesn't matter what you do with Sasquatch. You can choose not to search for him in the tunnel, not to complete his missions, or to not even reinforce his ass. It's not a problem if he dies at any point before you return to the tunnel. Just don't call for an extraction with the nuke armed. It's really that simple.

The option to begin rescuing refugees comes after completing the Julene Horn extraction. After that, Vienna Cho will begin talking about about refugees and badmouthing the army. You will probably get a chance to rescue two refugees before the next siege hits. Go ahead and rescue them and then keep them both alive during the siege.

Warning: this second siege - the one that comes after Julene Horn's extraction - is the only siege during the story part of the game where there is no high value asset to extract. You can extract two refugees on this one, and this one only. On subsequent sieges, you can only extract one refugee even if there are two refugees in your base and the HVA has already died. Furthermore the refugee who remains behind won't extract in the next siege, either, even if he's the only civilian there. He ends up basically becoming a permanent member of your community. This has happened to me twice, and both times, it caused the story progression to stop - meaning, Highroad never told me about the next HVA. So, if you get an opportunity to rescue two refugees in one siege cycle except for the one right after Julene Horn, don't do it. Just let the second one expire if you have to. It may set you back with Vienna Cho, but at least it won't break your game. Once you're finished with all of the HVA extractions - and you will know when this happens - then you can start rescuing two refugees. Only one will leave on the heli, but the game won't be broken.

From my own experience and from what I've been reading, the minimum number of refugees you need to rescue to get Cho to call you for help is seven. If you rescue two refugees in the first siege after Julene Horn and then one apiece in the next five, then you will have your seven at the same time you've completed the HVA extractions, and she should call you. I had to complete nine sieges because I failed to rescue one refugee, and then I didn't get an opportunity the next time. Cho still called me after I had rescued seven, even though one of them didn't extract.

Just to give you an idea, here's how my game went:
Siege 1 - Julene Horn extracted. Cho comes online afterward.
Siege 2 - Doghouse goes dark. 2 refugees rescued and extracted.
Siege 3 - 1 refugee rescued and extracted, HVA #2 extracted.
Siege 4 - 1 refugee rescued and extracted, HVA #3 extracted.
Siege 5 - 1 refugee rescued and extracted, HVA #4 extracted.
Siege 6 - Lost 1 refugee during rescue. HVA #5 extracted.
Siege 7 - No call fro Cho. HVA #6 extracted. Main story complete.
Siege 8 - 2 refugees rescued. 1 extracted, the other stayed at the base.
Siege 9 - Vienna Cho rescued, everyone extracted.

When you are going to rescue a refugee, take no chances. ALWAYS take a buddy with you. Be holding a throwable incendiary or explosive that you can toss immediately upon getting out of your vehicle. If you arrive and you see the refugee crouching down, pleading for help, and your plan is to melee the zombies off of him, you will lose the refugee, and you will deserve to hear the unpleasant things Cho will say about you.

By the same token, do what you need to do to keep them alive during the siege. You can have at least thirteen soldiers in your base - the five you start with, six stranded ones you can find from search and rescue, Julene's friend, and Sasquatch. This helps a lot. Always have your perimeter defenses ready, and plant some extra mines of your own. Call in for extra support if needed. When you're doing the final extraction, go all out and use up all your influence.

Please let me know via comments or PM if I made any mistakes, or missed anything important.

Just figured out how to get this even easier. Just follow the guide above until you get the choice to either arm or disarm the nuke. Choose to disarm the nuke and complete that mission. Then when you bring her back after disarming the nuke you will see the option to talk to sasquatch and set off the nuke. Go ahead and tell him to set it off. Since you disarmed the nuke it will not detonate. Just finish the siege and boom achievement unlocked.