I have a proof that I want to undergo peer review. I unfortunately am not affiliated with any university. How should I go about getting it reviewed and either rejected or published?
I don't publish in pure mathematics journals (I'm a computer science researcher) but at least in my field, there is no affiliation requirement in order to submit to peer-reviewed journals.
First a quick comment: you haven't told us the subject of your proof. I don't mean to sound negative or discouraging, but if it relates to a famous long-standing conjecture (Riemann hypothesis, twin prime conjecture, etc), there is very little chance your paper will be taken seriously, as these problems have been so well studied that the odds of an amateur resolving them in 6 pages is very very low. If you do find yourself in this situation, then surely along the way to proving the famous conjecture you have also developed new theory and partial results that are interesting in their own right -- I would focus on getting some of these published first.
The first step would be for you to identify a relevant journal, and read their submission instructions. All journals have very specific instructions about how your submission should be formatted and how it should be submitted/uploaded to them. Many have LaTeX templates you'll be able to use.
Next, the editor will determine whether to send the article out for peer review, or reject it immediately. To minimize the chances of outright rejection, be sure you structure and format your paper professionally. Reading accepted papers from the journal is surely the best way to learn the standards of form, but a few quick tips:
If your submission looks and smells like a professional paper, it will be submitted for peer review. The usefulness of those reviews is always uncertain, but with any luck you will get good feedback on how to improve your paper, even in the unfortunate even that the reviewers recommend rejection.
Ultimately, you have to write a complete paper around your proof and submit it to a journal. Hopefully, you already have an idea which journals might be appropriate for your paper.
There are probably many means to get feedback before, but these can't replace actually writing and submitting the paper. A tricky part might be to find and discuss related work (this is expected as part of the introduction of a paper), especially if you don't have access to anything equivalent to a university library. However, you don't necessarily have to be affiliated with an university to get access to its library (or the library of its mathematical institute).