For the English alphabet there's a site, FontSquirrel, that provides great free fonts for commercial use, handpicked and tested. I haven't found anything similar for Chinese fonts, the only very good free font and commercial-use OK is Google Noto, but I'd need something more "traditional-looking", a type of font you'd find in school books. If there isn't anything good for free, I am also OK with purchasing something, but I haven't found online neither a store for Chinese fonts.
You may want to try WenQuanYi bitmap fonts.
Unsurprisingly the Japanese do this better.
Hanazono have an excellent, free, font:
About Hanazono font
List of included characters
This font contains about 90,041 characters (and spaces) defined in ISO/IEC 10646 standard / the Unicode standard.
It's broken up into two files HanaMinA.ttf and HanaMinB.ttf and can be downloaded from Source Forge.
After having done several researches and having looked at your answers, I merged them and I finally got to a solution, that I will share.
Hanazono is undoubtedly the best free and commercial-use friendly font to display simplified Chinese yet its rendering is not perfect: it often gets blurry on desktop devices, while it's pretty sharp on mobile ones. So I found that the company Dynalab offers great S Chinese fonts for an affordable price (50 dollars), and this seems the best solution for people with a low budget. Their site is also available in English, which is great.
Since the topic also involved Japanese and Korean as we talked about it, I found out that there are some very, very wonderful fonts for these languages, all free and suitable for commercial use. For Japanese we got Aozora Mincho, which renders incredibly on all devices and screen densities, while for Hangul Naver itself developed Nanum Myeongjo, which is excellent on desktops and stunningly perfect on mobile devices.