I want to write a historical research paper I apologize if this isn't the place to post a question like this. Feel free to close it if that's the case.

So I want to write a historical research paper on a particular event in a past U.S. president's life. However, I'm not in academia. In fact I only have a high school education. Writing the paper isn't my problem, but I'm not sure what to do with it afterwards. Should I even try to publish it or just keep it to myself as a personal exercise?
There is no requirement to be in academia in order to be able to publish. You need to find a "suitable" journal for your work and figure out how to submit to them. Now a days it is mostly via an online form. They may ask for some background on you, such as affiliation, but that is more for informational purposes than anything. Independent Researcher is a valid "affiliation", generally. 

But you may want to get some advice from someone in the field if you have any doubts about the quality or relevance of your work. You may need to decide whether the academic or popular press is more appropriate for your work, and a pro may be able to help you with that. 

But, hopefully, your research took you through papers written in reputable journals. Those are the ones to consider first.