"exclaim against their own succession"

A quotation from Hamlet by William Shakespeare.

Hamlet is talking about the craze for a particular group of child actors that is sweeping the land. By doing the adults out of a job, Hamlet implies, these children will end up doing themselves a disservice by ruining their own chances of working when they grow up. Angel's use of the quotation is more literal, implying that some members of the aristocracy are actively questioning the system that sees prestige and wealth handed down through the generations.

 

HAMLET

What, are they children? who maintains 'em? how are
they escoted? Will they pursue the quality no
longer than they can sing? will they not say
afterwards, if they should grow themselves to common
players—as it is most like, if their means are no
better—their writers do them wrong, to make them
exclaim against their own succession?

Act II, Scene 2