The Outer Worlds

You unlock the achievement by going to the SMC facility in The Ambrose Intersection Mission and activating it. This is one of four possible endings in the game although, if you choose the right three endings, you can get all ten of the DLC's achievements.

The four endings are:

There are two ways to get the achievement. The first way to get the achievement is along the way towards getting the following:
If you need the Postman achievement, before acquiring Olivia's journal in the Red Tape of Command mission, go to the Ambrose Mansion and kill Minnie. This triggers Lucien's appearance when you return to the ship and activate the navigator. The conversation is not binding so you might as well be noncommittal although, eventually, you will take his side.

Conclude the Red Tape of Command mission by acquiring the journal. It's easiest if you enlist PAM, the robot's help--having Parvati as a companion will make this easy because of her engineering ability. PAM, whom you encounter right inside the entrance to the facility, will lead you to the final elevator and assist greatly in eliminating hostiles. You do not need her. You can follow the objective marker and eliminate the hostiles yourself, but it's a fairly convoluted trip with quite a few hostiles.

Concluding the Red Tape mission, starts the Ambrose Intersection mission. In any dialog with Lucien and Olivia either take Lucien's side or be noncommittal. The conversations are non-binding though. The outcome is determined by your actions not the conversational choices. The easiest way to conduct the conversation is to be noncommittal. Proceed to the mission's objective marker on the SMC facility ignoring the second mission's objective marker on the Ambrose Mansion.

You will meet up with Olivia. There's no need to converse with her, Choose the option to attack her directly. Kill her and loot her body to get the reactor activation key. Use it to power up the reactor and the achievement will pop. Alternatively, there is an option to kill Olivia by throwing her in the reactor. This turns out to be the simpler option. You no longer need the key to activate the reactor and you avoid combat--a short cut scene eliminates Olivia following the dialog choice. Following this cut scene, power up the reactor to unlock the achievement.

To get the Postman achievement, when you return to your ship, talk to ADA & have her connect you to Lucien. Tell Lucien you have the journal and will give it to him. He arranges a dead drop on Byzantium. Go to Byzantium (either landing pad works). Proceed to the dead drop and place the journal in it. Talk to the man standing nearby. The Postman achievement will unlock. If it is your first time completing the DLC with any ending, you will also unlock the Peril on Gorgon achievement.

If you do not need The Postman achievement and/or want to see all endings, use the second way to get the achievement. Don't kill Minnie. Acquire the journal during the Red Tape of Command mission. In the dialogs with Minnie and Olivia, be noncommittal. Your choices in the conversation don't matter because the outcome will be determined by your actions.

When you have control of your character for the Ambrose Intersection mission, proceed to the mission's objective marker on the facility, ignoring the second mission's objective marker on the Ambrose Mansion. You will meet up with Olivia. There's no need to converse with her, Choose the option to attack her directly. Kill her and loot her body to get the reactor activation key. Use it to power up the reactor and the achievement will pop. Alternatively, there is an option to kill Olivia by throwing her in the reactor. This turns out to be the simpler option. You no longer need the key to activate the reactor and you avoid combat--a short cut scene eliminates Olivia following the dialog choice. Following this cut scene, power up the reactor to unlock the achievement.

Although powering up the reactor unlocks the achievement it does not complete Minnie's ending. To do so, return to the mansion and converse with Minnie. This will finish the story. If it is the first time you have completed the DLC with any ending, you will unlock:

Thanks to @Thall802 regarding the conversational option to throw Olivia in the reactor.