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So, is this how Nurgle and Tzeentch work? Did I get this right?

Tzeentch is the God of Hope, and people who have everything to gain. A guy who wants to be King would serve Tzeentch.

Nurgle, on the other hand, is the God of Despair, and people who have nothing to lose. The followers of Nurgle do whatever they want, because things can't get any worse for them. They live free from consequences, rules and regulations.

Is this right?

From what i know, you got it pretty much right

"In Nurgle's case, the source of power is the mortal fear of inevitable death and disease, and their unconscious response to that fear, which is the motivating power that drives the lives of mortals."(as seen on lexicanum)

the difference between the followers (40k twitter, check in in the eye of terror) of those two gods also explain the rivalry between them

"Nurgle's main enemy is Tzeentch, the Lord of Change, as the two draw their power from opposing sources. Tzeentch is hope and changing fortune, while Nurgle is defiance born of despair and hopelessness."(again, lexicanum)

Tzeentch is knowledge and hope.

A would be king would pray to Tzeentch for knowledge that would give him power. This might be done with a complicated scheme to depose the existing king and get himself crowned instead.
The fall of the nine kings in Lord of the Rings is how Tzeentch would work.