This is for completing the infiltrate the cult mission, there is some requirements for this and some recommendations.
1) This is a mission that seems to appear only in the late days (5 and beyond), so your better option is to keep corruption controlled over the continents.
2) Mira is the member to bring (maybe others work too), since she has some positive outcomes and CAN infiltrate the cult.
3) There is a pre-mission, on the day before called Cult of Gargos you need to complete. Don't forget to bring Mira, before this mission send her to go near the guards and, after it, send her to infiltrate the cult. I think if you don't have an orange dialog, you won't be able to infiltrate it.
4) On the Infiltrate the Cult, pre-battle, you should choose the option to answer the cult leader and after the fight choose the option Search the Cultists' lair or Do a Cleansing Ritual (Thanks https://www.trueachievements.com/Nonnenteufel1.htm
5) I've only seen these missions, after I completed the quests Temple Under Siege and Deal with the Devil, in this order. Also, make Mira your first option on these, because she has some positive outcomes. On the first mission, choose to release the cultists, which will unlock the second quest. And on the second quest, choose the orange option marked for mira. It was brought to my attention by SiegfriedX
that you can unlock the achievement without playing them in the same run. I'm still holding confirmation if they are real pre-requisites that can be done in other play-throughs.
6) What I actually did was start a new play-through and skipped turns until I got one of the missions of interest in the first days. If I could not get it I just restarted.
So, in summary, always bring Mira, complete the quests Temple Under Siege (release cultists) and Deal with the Devil, wait finishing all possible missions, while postponing the arrival of gargos. In the end, complete the Cult or Gargos (Infiltrate the Cult) and Infiltrate the Cult (Search the Cult's Lair), again with Mira.
Per comments, another possible character than can be used is Shadow Jago, if you don't have Mira, he is a possibility.
If you don't have Mira or Shadow Jago, one possibility is to defeat him (the real one not the mimic), during in your play-through. In this case, he will offer to join and you can use him, during this play-through, even if you don't own it.
This is what my memory was capable to retain. If you have corrections or more details, please put in the comments.
UPDATE: Exe the Hero
pointed out that you can accomplish this with Hisako, playerforty4
pointed out that you can use Orchid, Fummm
pointed out that you can use Sadira and, finally, Fejer
said hed it with Kan-ra.
Thanks to dieotze74
for some tips and corrections.