The Scholarly Kitchen recently ran a post and video showing how ink is made.
“The post is dedicated to all craftspeople — photographers, editors, writers, designers, papermakers, typographers, videographers, and audio engineers — being asked to stretch and change by external forces beyond their control.”
(Note that The Troy Book Makers is a digital printer, so the interiors of our books use toner as opposed to ink, but this process could still apply to 4-color covers that are printed offset.)