It's pretty simple. I'm a developer for a big media company that has buzzfeed-style quizzes all over the web. For what it's worth, here's our model.
Start with the outcomes/answers. This would be the presidential candidates. Ie, Clinton, Trump, Sanders. Once you've defined those...
Start writing questions. Each question will give weighted points towards a particular outcome. I'm not particularly political, so I'll keep it simple and could be totally wrong here...
"How big should government be?"
government should have a strong role to directly address the challenges faced by the American people, especially those which pertain to the less privileged in American society. (2 points for Clinton, 1 points for Sanders, 0 points for Trump)
government’s only legitimate role is to protect individual rights to life, liberty and property, and not abrogate these rights. It is right to have laws against murder, assault, rape and theft, but actions that do not intrude on the rights of others should not be restricted. (2 points for Sanders, 1 point for Trump, 0 points for Clinton)
government is to protect individuals' rights and that individuals and society as a whole are better off when the government is involved as little as possible. (2 points Trump, 1 point Sanders, 0 points Clinton)
At the end of the quiz, add up the points, and whichever candidate has the most points would be the one you'd recommend they vote for. And if there is a tie, you can handle it in a couple of ways. Either present them with "You could either vote for [candidate A] or [candidate B]" - meaning, just present the user with a tie. And/or if you don't want to just present with a tie, you could present them with a bonus question (or the option to answer another question to break the tie, if they want to) that serves as the tie breaker. This could be a question that is a fundamental difference between every pair of candidates (one for Hillary vs Trump, Trump vs Sanders, Sanders vs Hillary).
To code this out on the front end is not a big deal. Probably a few days worth of solid work with a clear UI/layout to work from. And it could be 100% front-end (no server-side work needed). A week from start to finish just to give some pad room. But the actual time to develop it would depend on what all functionality you want on the site, so you'd have to clarify that.