The Codex of the Underworld is a book given to Zagreus by Achilles, containing information on the characters, creatures, and objects that exist in the Underworld and beyond. Achilles hoped that by giving the book to Zagreus, he could encourage the prince's love of history to grow, rather than only teaching him how to fight.
The Codex functions as a semi-magical record of Achilles' own travels and experience, both on the surface (e.g. with the Olympian Gods) and in the Underworld; therefore, it is not unbiased or objective, but is written from Achilles' point of view, and references his personal opinions, relationships, history, and understanding. As Zagreus encounters, gathers, speaks to, or slays the subjects of the Codex, he is able to access Achilles' knowledge more deeply, and the entries become more complete.
The Codex also serves as a reference of how many bottles of Nectar and Ambrosia have been gifted to certain characters, via a row of hearts in the top right corner of their entries. There are ten hearts in total, but for most characters, the highest heart levels are currently locked and unavailable.
Though the Codex is obtainable at any time after Zagreus' third escape attempt, Achilles may not hand it over right away - multiple conversations may be required to trigger the correct event. The Codex cannot be opened prior to obtaining it.
The Codex currently contains the following headers and entries (click to expand; click through to individual pages to read):
The sections of the Codex are ordered in different ways.
Chthonic Gods - Hades is first, as the title character of the game and lord of the Underworld; he's followed by Zagreus, the prince and protagonist, and Nyx, in order of eminence. Charon follows the royal trio as the leading psychopomp and therefore necessary for the Underworld to function. Hypnos (followed by his brother, Thanatos) and Megaera (followed by her sisters, Alecto and Tisiphone, in alphabetical order) are likely listed in order of when the player would be expected to encounter them. Chaos, as the shadowy and oft-forgotten primordial deity that they are, is placed last.
Olympian Gods - Zeus is at the top of this list, as lord of Olympus. He's followed by his older brother and second-in-command Poseidon. After that, Athena is listed as the first of the "younger generation" of Olympians likely because she is the first Olympian the player meets, and the most outwardly supportive of Zagreus' efforts. After this, Zeus' children (and Aphrodite, whose lineage isn't clearly stated in the game's canon) are listed, seemingly in alphabetical order but for the placement of Artemis above Ares. This may have to do with how many runs into the game these gods can be met, or may not have any particular significance. Dionysus, Hermes, and Demeter are presented in the order in which they were added to the game.
Others of Note - These characters are listed, broadly speaking, in the order you might meet them after reviving at the Pool of Styx. Achilles and Orpheus are present in the House; Sisyphus is present in Tartarus; Eurydice in Asphodel; Patroclus, Theseus, and Asterius in Elysium; and Cerberus in Styx. It's unclear why Dusa and Skelly are at the bottom of this list.
The Underworld - These places are nearly listed in order from deepest to highest within the Underworld. However, the House takes the topmost position as the most important to the Underworld's administration.
Perilous Foes - Enemies are grouped first by what biome they initially appear in, and then by their "subtypes". This leads to some initially odd-looking ordering, such as Voidstones being placed in the part of this section dedicated to Tartarus enemies. However, Voidstones are a subcategory of Brimstone, and so are listed after them, transitioning from Tartarus to Asphodel foes. Subsets of Bloodless also appear in Tartarus as mini-bosses, but are listed in the Asphodel section as subcategories of Bloodless; as they also appear more often in Asphodel than in Tartarus, this is appropriate. Of the Exalted subclasses, it's interesting to note that they're listed in the same order as the weapons in the "Infernal Arms" category - Sword, Spear, Bow, Shield.
Infernal Arms - These are nearly listed in the order they're displayed in the weapons room at the House.
Artifacts - Darkness, as the most basic, thematic, and pervasive item in the game, is listed first. It's followed by "gold laurel" or run-limited power-ups that Zagreus can use to strengthen himself over the course of a run, but which he loses when he dies. Beginning with Chthonic Keys, the next section is dedicated to "blue laurel" or meta-run items that stay in Zagreus' inventory after he dies. The Red Onion, as a sort of insignificant spoof item limited to Erebus rooms, is listed last.
River Denizens - The fish in this list are ordered by their environment from deepest to highest (Chaos > Tartarus > Asphodel > Elysium > Styx) and then by rarity (Common > Rare > Legendary).
Fables - The stories of the Chthonic Companions are listed in an at first strange manner. The first two, Battie and Mort are presented in alphabetical order (both by their names and the names of their previous owners) and above the others as gifts from Chthonic Gods. They are followed by Fidi, Shady, and Rib, which are organized alphabetically by the name of their previous owner - Dusa, Sisyphus, and Skelly respectively - and below the first two as gifts from non-deities.