Fia's questline begins in Roundtable Hold and eventually connects to Deeproot Depths, the Cursemark of Death and Lichdragon Fortissax.
Roundtable Hold
Speak with Fia and continue her dialogue as the story advances. One important middle step involves the Weathered Dagger and D, Hunter of the Dead.
The Cursemark connection
Fia's route later needs the Cursemark of Death, which is connected to progress through Ranni's questline and the Divine Tower of Liurnia.
Deeproot Depths
Eventually Fia relocates to Deeproot Depths. Reaching her there is a major checkpoint, so use a note or checklist if you leave the quest for a long time.
Prince of Death's Throne
Continue Fia's dialogue at the Prince of Death's Throne and provide the required quest item when the option becomes available.
Fortissax encounter
The later part of the quest opens an optional encounter with Lichdragon Fortissax and leads to the quest's ending-related reward.
Do not mix the routes
Fia and Ranni overlap, but they are separate questlines. Track their checkpoints independently so one route does not become confusing when you pause the other.
Reference: GameSpot Fia quest guide.