So I just got the skill that lets me start a pirate empire, but I can't figure out how to use it/do it.

Have a good relations with any governor (except governor of capital star systems !) & one day he will tell you that he don't like current goverment (or something like that - don't remember) . Say "Yes" & here we are! Our own capital system.

Other systems you must conquer. There is no diplomatic takes.