To get K2 at the end of chapter 5, you'll need to be placed in k4 initially (get high slave points in the jail), and then pick Scarlett instead of Junko when you're on the plane, so that you'll meet Julliet instead of Dominque. Julliet will demote you to K3. Then, when you meet your sister, don't align with her or give her info about Katsumi. Have her piss on you when she visits you in the K3 area, and completely fail your day (Pick all the wrong answers about Alessandra/Celeste, etc) so that Chanel gives you a terrible recommendation.
To get K1* (It's not technically K1, as there's no return from actual k1) at the end of chapter 6, you'll need to follow everything above, except Olivia also has to be evil. Completely fail your next day, pick all evil choices for Olivia and then you'll be demoted, but you'll have a chance to use your Kpass if you want.