Kodansha is the term given to the higher black belt ranks. Most often this is regarded as ranks at a minimum of 4° or higher.
Achievement of Yudansha ranks: Shodan - Sandan are fairly straightforward with a specified time in grade and a specified Kata requirement.
Beginning at Yodan the time in Grade remains specified but the Kata Requirement becomes varied.
The first thing to note is not all Katas that exist in Judo are recognized Kodokan Katas; because of this, it is recommended to study the recognized Katas first before moving onto those outside of official Kodokan Judo Catalog.
The recommended order is as follows:
Shodan: Nage No Kata
Nidan: Katame No Kata
Sandan: Ju No Kata
Yodan: Kime No Kata & Goshinjitsu
Godan: Koshiki No Kata & Itsutsu No Kata
These are all Kodokan Official Kata
Beyond Godan the Study of Non-Kodokan official Kata can be studied. These can be studied before this level too but it's unlikely they'll apply toward promotion due to their absence from the Kodokan Curriculum.