When writing an essay, how do I speak of someone who has a Ph.D?

Call her Dr Mary Sue. She has a doctorate degree, and she's done a LOT of work to get that, so she is entitled to be called a doctor. I know what it involves so she deserves that! Do not mention the PhD at all unless you need to, to point out that she is not a medical doctor but an academic doctor. Mary Sue PhD is also possible but in normal speech that just sounds silly.

Take when I was at university. In British universities, only the top members of staff are professors who you would call "Professor X". The rest are readers and lecturers, who almost all have PhDs too (we had just one who didn't). So if I were talking to another student after a lecture that confused me, I might say something like "Did you understand what Dr Mary Sue was talking about this morning?"

Mary Sue sounds like her first name, though. You would, in fact, use Dr with her last name and call her Dr Smith or Dr Brown or whatever. Or Dr Mary Sue Smith or Dr Mary Sue Brown if you need to.

Any one with a Ph. D is a doctor.

Dr. Mary Sue

or if you want to keep the Ph. D in the name write it:

Mary Sue, Ph. D

Refer to them as "Dr. So and so." You can add a footnote explaining their academic credentials.

dr dre he got a phd in hiphop

You TALK to them the same way as you talk to anyone else. If they insist you refer to the degree at all, they are not worth talking to.

Be yourself.