Finding an advisor without enrolling into a university

First of all, I wouldn't seek an advisor. Under your circumstances, you shouldn't expect a professor to assume any sort of responsibility for you. That said, you might be able to find someone who is willing to help you. Here are two ideas:

1. Find a graduate course that you are interested in, and ask the professor if you can sit in on it. The professor might well say no, but they might say yes. If they say yes, then work hard, do all the homework and do it well, and ask good questions. When I was applying to graduate school, I was very fortunate to have a professor who was willing to do this for me.

2. Attend a scientific conference. Go to talks, and talk to people afterwards. Ask questions. If you have an idea for how to make progress on a question of interest to one of the speakers, you may be able to strike up a collaboration.

In either case, you could ask for a rec letter later.

Finally, at least in the US, it's common to apply to PhD programs with only an undergraduate degree in the same field, and to get letters of recommendation from your undergrad professors. So, if you recently completed an undergrad program in CS, and you made a strong impression on your professors, then you might try simply applying to PhD programs now.

It's not impossible, but it seems hard. The problem is that faculty members at research universities typically get credit for advising students who are enrolled in their programs; in turn, universities give them credit because students pay tuition, which supports the salaries of the faculty.

The other way that faculty members get credit is by authoring (and co-authoring) papers. If you want to convince an academic (or I guess someone in industry, although I don't know much about how industry works - I would guess that they would have even less slack for "side projects" such as advising an external student), I'd say you need to make a good case that:

• working with you is likely to lead to publications on which they can be listed as a co-author;
• you won't take up too much of their time — that is, the time they invest in advising you will be commensurate with the number and impact of papers that you co-author with them. Among other things, you'll need to convince them that you're well-prepared to start on a research topic, and that you're capable of working more independently than is typical for a graduate student. The "papers/advising hour" ratio might need to be higher than for a typical student (since you won't be paying tuition/countable for credit in the usual university metrics).

As for "how do I go about contacting a potential advisor?", the answer is pretty much the same as for students looking for a graduate supervisor. Find people who are working in your desired field; consider contacting their students (or former students) to find out what they're like to work with; and send them a brief e-mail outlining your goals and persuasively explaining the points above.