There's actually three questions here, so let's take a look at each:
- The question you asked: How do I publish as a non-academic?
- The question you implied: How do I publish a paper refuting another?
- The question you should ask: Is it worth publishing a refutation of a master's thesis?
1. How do I publish as a non-academic?
Assuming you've got a paper you want to publish, the first step is to determine where to publish. Generally, your options are at a conference, or in a journal. Which direction to go depends a bit on your field.
- Conferences are generally the place to publish (relatively) small, incremental works. Presenting your paper at a conference can generate valuable face-to-face discussion to inform your next steps. My field (computer science) skews towards conference publishing.
- Journals are better for larger works with significant advances or particularly-reliable evidence. They're also better if you're not looking to travel to present your work. Older, more stable fields tend to do more journal-based publishing (e.g. I believe Physics tends to be journal-focused, though I'm not an expert).
Once you've determined your preferred medium, you'll need to pick out a specific conference/journal. This is entirely field-dependent, and your best bet is to find someone who has published in your area and ask advice. You can also generally get an idea of what a particular conference/journal is looking for by reading their call(s) for papers.
Once you've selected your venue, you'll need to read their publishing guidelines. Typically there will be a required style/format, usually accompanied by a template Word or LaTeX file. There will also be instructions on the actual submission process, typically run through a 3rd-party submission website. The actual submission is usually as simple as creating an account, uploading your files, and filling in your title, author list, and abstract.
(note when I say "usually," what I really mean is "in computer science")
2. How do I publish a paper refuting another?
This shouldn't change much about the publishing process, other than the fact you should strongly consider publishing in the same venue as the original paper.
The writing of the refutation paper is probably going to look a little different than a normal paper, but I can't speak with authority on the subject as I've not written such a paper before. I'd just point out that unless you believe the original author was behaving maliciously/unethically, you should look for a tone that redirects more than refutes.
e.g. instead of saying "Author OP said X, but that's wrong because Einstein et al. said Y," synthesize the two: "Author OP investigated question A, resulting in claim X. However, this is inconsistent with Einstein et al's finding, Y. Further work is needed, modifying OP's approach in the following ways, etc."
3. Is it worth publishing a refutation of a master's thesis?
This is the question most of the answers here address. And I agree with them, the answer is most probably a simple "no."
A couple sub-questions to find your answer:
- Was the thesis published in a journal or conference proceedings?
- Did the author of the thesis go on to publish other works based on the thesis (a Google Scholar search or similar would help answer this)?
- Does your field, specifically, consider masters theses to be meaningful, significant contributions to its body of academic work?
If the answer to any of these is "yes," then it may be worth going ahead with your refutation. In particular, if the thesis was published in a reputable, widely-read/attended venue, then there's a very good chance the publishers would be interested in a paper that challenges or recontextualizes the original paper.
On the other hand, there's a very good chance that the answers to those questions are all "no." And in that case, there's very little chance any academic community will care enough to publish your refutation. Writing papers is a lot of work. Reviewing papers is a lot of work. Publishing papers is... well, still work, if a bit easier in the digital world. Instead of focusing effort on correcting a paper that was only seen by three professors at one institution for some master's student's final project, focus on ways to contribute your own expertise to the field directly.