How do the round systems work?

Each chapter (to keep using that terminology) has six rounds, except for the first, which starts with a draft pick (everyone picks at once) and then three minion rounds.

The fourth round of each chapter thereafter is a carrousel round where you can pick a champion in order of lowest to highest score thereafter. After the carrousel round, there is a PVP match in the same round. Comparing the screenshots below, the chapter looks like this after the carrousel part has finished (opponent's name has been blurred out):

Round six of every chapter except for the first is a monster round (Krugs, wolfs, Raptors, a dragon, Mechs, Rift Harold, the Elder Dragon, respectively).

All the other rounds are PVP.

To illustrate this, I took screenshots starting at the first chapter with PVP and ending at round six. _{Not showing rounds thereafter because all other players were eliminated by then. ;)}

Source: Rankboost (note that they use zero-based numbering and as indicated in a comment, they leave the chapters out altogether)