The name doesn't appear in any of the canonical episodes of the cartoon series. The narrator mentions it in the noncanonical "When Mother Brain Rules" clips episode, and it supposedly has its origins in the production bible.
The Nintendo Power article previewing the cartoon series refers to them as the League of Darkness (possibly an early name that was discarded), which was eventually used in the comic book continuity.
It is unclear when the Forces of Chaos was first formed and unknown why the individual members joined. Depending on the definition (see below), it likely was formed at least seven years prior to the beginning of Season 1.
Since the term "Forces of Chaos" is never spoken in a canonical episode, it is unclear whether it encompasses just these individuals or Mother Brain's entire army. "When Mother Brain Rules" has the narrator seemingly indicate the latter.
The hierarchy of the Forces of Chaos is never made clear in the series.
Mother Brain is obviously the leader.
Dr. Wily is never shown interacting with King Hippo and Eggplant Wizard outside of Mother Brain's presence, though it can be deduced that Mother Brain would place him in charge over the others due to his intellect (when he makes an appearance).
King Hippo would likely claim rank over Eggplant Wizard by sheer brute force, in addition to the latter's timid personality.