Chthonic Companion's are required through the same method as Keepsakes, but become available when your relationship with specific characters grow to a larger level and by gifting them Ambrosia instead of Nectar. When you give Nectar to a character, their relationship grows and on the first exchange you will receive their keepsake.
As you progress in your runs, make sure you talk to each character every chance you get and when you have nectar available, give it to them as a gift to boost their relationship status. Eventually in the codex, you'll see their status shows a heart with a Keyhole. If you have Ambrosia (which is mainly unlocked during runs on the Elysium boss fight and through fulfilling prophecies), you will be given the option to gift it to them. A unique dialogue will play indicating that the characters have grown extremely close and you will be rewarded with their Chthonic Companion.
The list of Characters this is possible with is:
Companions themselves are like Summons. They each offer a different ability you can call upon during a run. You start out with only being able to use them once per run but with added upgrading at the Display Case in the Courtyard, you'll be able to use them multiple times throughout each run.