Using a figure from a journal article on a personal website I'm writing an article on a personal website and in some of the reference material that I'm already citing, there are a few good figures that I'd like to include in the article.

This is probably a silly question but I'm wondering if this is allowed under copyright laws. Specifically, if I cite the original paper/source, would I be violating any copyright laws? Should I be contacting the authors of the paper for permission?
It's probably a copyright violation if you don't get permission. It's also probably covered under Fair Use laws in the US, but Fair Use is a defense against a lawsuit not a thing that prevents people from suing you, so you might be subject to a whole range of legal trouble starting with a DMCA takedown request and extending through a to lawsuit. You might win all these situations, but you might be out tens or hundreds of thousands of dollars. 

If you want to use it with permission, you should contact the copyright holder. Find the copyright statement on the first page of the journal article or in the front matter of the book, e.g. "©1995 Elsevier" or whoever, and contact them. If it's a big publisher, they probably have a use, licensing, or copyright clearance web page for requesting permission to reproduce small parts of works they own. In many if not most cases, the author is the wrong person to contact since they likely transferred the copyright to the publisher as part of their publication agreement. Usually, especially for academic works, the license to use a figure for academic/commentary purposes will be free.

If the copyright is held by a university or the author(s) personally, you might start with an author. They might be willing to give you permission directly if they can.