If you have a doctorate in pol phil from a strong program (say, Arizona, Harvard, Princeton, Duke, Toronto, Brown, Oxford, ANU-- yes, I know that doesn't match Leiter's rankings, but I don't understand those rankings in pol phil) and a JD from a top-10 university, you certainly have a chance at serious consideration by top programs. Depends on the substance of your dissertation, and as always 1-2 top publications could make a huge difference. But that's a combination that will be very attractive. The law degree could credibly give you a public law AOC (we don't talk about AOS and AOC in poli sci, but we have equivalents). And there are still plenty of political theorists (not me) who have philosopher envy the way our formal and quantitative colleagues have econ envy, and think that a smart person coming from the other, higher-status, discipline is to be actively preferred. Certainly a department like Brown, Princeton, or Stanford-- all of which have philosophers on their faculty-- will be happy to see your application.
But you won't be competitive for most jobs below the top tier. You probably can't teach history of political thought in any serious way, and for a department with few theorists that's an important teaching need. And not knowing much poli sci makes you a less attractive hire and colleague in a department that only has 1 or 2 theorists-- the theory group's not large enough to be your whole community, and everyone else will think you won't have anything to talk with them about. And the non-theorists who will be hiring you in a mid-level program won't be in a position to assess your work, won't know the names of your advisors, etc.
"I have a JD, and would be happy to teach a range of courses covering American Government, Constitutional Law, Comparative Constitutional Law, Electoral Systems, etc. Not sure what is the best way to demonstrate 'competence.' Is the JD relevant? Are publications in one's AOC required? Coursework? Ability to answer 'how would you teach X' questions compellingly?"
Well, there's no reason to think you can teach American Government, and it tends to annoy specialists in any area when someone without training in it says they could just pick up the textbook and teach. The JD is relevant to Con Law/ comparative con law, and if you had serious law school work in elections law then that's relevant, but still doesn't qualify you to teach elections to political scientists' satisfaction.
Publications are sufficient to demonstrate a credible AOC. A law review article could do the necessary work (not a Note, though). Coursework is not sufficient. A comprehensive exam is usually the minimum necessary, and having TA'ed or taught a relevant course already complements that. But if you have courses from law and philosophy that overlap and reinforce in some compelling way (elections law and democratic theory, say), you could make a case. The ability to answer "how would you teach?" is necessary, but you won't get far enough to get asked that question if you don't have some of the other pieces in place.