I studied Greek and Roman politics through political philosophy courses as part of my political science program. These are some of the sources that helped me understand their systems. However, you might be better served by asking faculty in other departments or the subject-matter librarian.
Start with the Athenian Constitution. It's what it says in the title: a discussion of the constitution of Athens. It may have been written by one of Aristotle's students, or Aristotle himself. Aristotle had a long-standing interest in researching constitutions and the structure of governments (what we would today call "regimes").
Another excellent source is the Landmark Thucydides. This is translation of Thucydides Peloponnesian War with plenty of annotations. In particular, the appendices offer short, but insightful, forays into Greek political life. The stories within the text are also illustrative.
There are a large number of textbooks which include discussions of the structure of Roman government. However, none of that seemed very "alive" to me until I read Discourses on Livy. This is Machiavelli's translations of some works by Roman historian Titus Livy. It isn't merely a translation; Machiavelli also provides explanations and insights into Roman political life. Be forewarned that it isn't without bias. Machiavelli is a pro-republican figure and he sees both dangers and hope for democracy in the Roman system.
If you are studying political science in a university, you might be better served by asking the faculty of another department. Ideally you could ask your department of the Classics (or Greek or Latin), if you have one. If not, then a department of history or philosophy should be able to point you in the right direction.
Failing everything else, try asking a subject-matter librarian in your university. Their entire job is to identify and locate key sources of research.