Update: Several users have said in the comments that you can make a manual save before talking to the girl in the Fablesphere during Syanna's path. You can load this save later to get the Kling of the Clink achievement. This is much quicker than reloading from the start of Night of the Long Fangs.
This solution contains story-related spoilers.
This achievement is unlocked at the very end of the story. The first step is to make a manual save when you reach the "Night of the Long Fangs" main quest. At the beginning of this quest, you can decide to either go to Orianna who will reveal the Unseen Elder's location, or go to Damien who will tell you Syanna's location. If you choose to find Syanna, you must talk to the girl with flint and take the ribbon from her when you enter the Fablesphere during the quest "Beyond Hill And Dale" (the Fablesphere is the dream world with the fictional characters in). This is an optional objective. If you do not complete that objective, it will result in Syanna being killed by Dettlaff, which voids the achievement. Alternatively, if you selected the Unseen Elder path during Night of the Long Fangs quest, simply proceed to the end of that quest and you will get the achievement after killing Dettlaff and proceeding to the medal ceremony in Beauclair. It will also unlock at that point if you chose Syanna's path and she was not killed by Dettlaff.
I recommend saving at the start of Night of the Long Fangs quest so you can reload later and choose the other path. The Kling of the Clink achievement requires you to choose Syanna's path and to avoid talking to the girl with flint in the Fablesphere, which results in Syanna's death. It is not possible to get both achievements in a single run. If you do not have an earlier save to return to, you would need to play through the entire DLC again from the beginning.