I would like to write a brief academic paper that builds upon some existing research that I have found, but introduces some (what I think are)new ideas. I would also like to open source some code as part of what I have researched and built. Is it possible to write a paper without an affiliation to a university and how would one go about this?
Yes. There is no academic requirement for publishing research papers. Anyone can submit a paper for the peer-review process at most journals. However, as an outsider it would be better if you collaborate with someone familiar with the journal and the research publication process. There are lots of esoteric rules that the journals take very seriously (e.g., formatting, common research terms, language, and tone) that is impossible to know unless you have experience with the publication process. You should also note that each journal is different in terms of what they specialize in, the types of topics that publish, and their target readership. The more inside knowledge you have about the journals, the quicker you can get your paper approve for publication.
You would send the paper to the most appropriate journals or conferences - just like you did whilst at university. Most conferences welcome "industry" papers. If you are an ACM or IEEE member, then you can get an email address as part of your membership package and that gives you a more "professional" image.<p>I know of several much published "researchers" who only show their consulting company as their affiliation. That is sufficient proof for me that it is possible.<p>If none of the foregoing feels right for your specific situation, then have you considered contacting members of your former university to act as co-authors?
Conferences are vastly better than journals for getting the word out. Nobody reads journal papers unless they get word-of-mouth momentum - which often happens at conferences.