What is the irony of an assassination to stop a dictatorship and the rule of the triumvirs?
In an ideal sense, the purpose for the assassination of a political leader, particularly a leader who is perceived to be dangerous to the social and political order (a tyrant), is to remove that danger and install a friendlier government or governmental system in its place. Realistically, assassinations are the tool of those who seek to gain power for themselves by eliminating their competition.
In Shakespeare's Julius Caesar, the assassination of the title character embodies the idea in the latter sense much more than the former. The conspirators against Julius Caesar, speicifically Brutus and Cassius, are motivated not nearly as much by the protection of the social and political order as they are by the desire to fulfill their own political ambitions - the installment of a new political apparatus with them in the seats of power.
This view of assassination expresses the irony inherent in such an act. Most would assume, at least in an ideal sense, that those who seek to assassinate a leader they perceive to be tyrannical do so out of a concern for the social order. In Julius Caesar, however, the goal of the conspirators might be to stop a dictatorship, but not out of republican virtue; their goal is to remove one dictator in favor of another. The act of assassination, while it kills the tyrant, ironically does not stop the dictatorship; the dictatorship merely changes hands. The people, often perceived as the benefactors of such an act, actually do not benefit from it.