If  you have the proper visa to do so, then there is no problem. Just make sure that your visa allows it.

Unless the person hiring you gets you the visa and takes you on as a legal employee, then no, it is not legal.

Work visas only allow you to work for the company that hired you.  Any other stream of income is technically illegal and can get your visa revoked if someone really takes a disliking to you.

Your work visa authorizes you to work for your employer only, private tutoring for a fee (or free) is not cover at all by your work permit (which is tied to your employer's name), thus illegal.  All you need is a student or parent upset at you, and a phone call to the Police to put you out of business, loose your visa and get expel from China, after paying a fine and spend a few days in jail.

Now, that is the letter of the Law.  it is up to you to accept the risk or not.

Technically is it illegal. Yes. Is it enforced or are you going to get in trouble or does anyone care?  No.  Not unless you're out causing other problems that involve you being arrested.

Every single foreigner I've met in China tutors on the side or at least has at some point. Every single one, no exceptions.  Only once did I see someone get in trouble for it but the real reason they got in trouble is because they got arrested for another matter and while at the police station they started asking why the person has money in his bank account if he doesn't have a work visa.  In the end he had to leave for one month and he was back the following month on a proper visa.

Think if it this way....would you work say they are fine with it if there was a substinal risk of you getting in trouble/deported over it?  Doubtfull

To answer your other question. If you aren't already working at a teaching or studying in a University where it's quite easy to find people asking you for personal tutoring I suggest making some simple flyers and putting them up on light posts around town or leaving them in small shops.  I've done this in the past and it always worked out well.  

My advice to is that it's fine (although TECHNICALLY illegal). It's a great way to earn some extra cash, help people out in the process, and also meet a lot of really good people. Some of my best friends are people I met through tutoring.

On a side note, I know a couple police officers in China (even tutor one of their daughters) and I've asked them about the whole private tutoring thing. And they agree 100%, unless your out causing other problems the police don't care if your hanging out with people and teaching them English in exchanged for a few hundred RMB an hour. Not a big deal nor is it a priority for law enforcement, especially since in the end you're helping people which in the long run helps the country of China.

Oh one more thing... I recommended putting up fliers instead of posting ads online because when you post ads online you're just going to get calls from learning centers/3rd party companies offering you a tutoring job.  It's very rare you'll get calls from invidual people. 

This is why I go the route of putting up fliers myself instead of posting an ad online.

if you can, I suggest having the flier in both Chinese and English on the same side. Even if someone can't read English the flier being in English tends to grab peoples attention but they can still read it since you have Chinese on there as well. Doesn't need to be anything fancy, my fliers were usually something very simple like "Private one on one English lessons with Native English speaker from America. For more information call: xxxxx "