Several papers have a helpful classified's section.
To secure an internship aboard, it is usually necessary to find a company that sets-up the internship. Internships are commonly set-up through a university for its students. Some agencies require a fee to secure a placement. Such opportunities include:
An excellent resource for is the HHS Center in Beijing. They are a group of experienced Chinese and foreign experts that match foreign youth with opportunities within the city for free. They work with variety of Chinese partners nationwide as well as with professional and licensed international agencies. Find out more about the application process on their site, HHS Application.
If you know anyone in China- get in touch! It's not necessarily what you know, but who you know. Friends and family, or any contact you have in your desired business may know of a position. There are also professional social networks like Linkedin which allows for people to make business contacts.
If hired through an agency, the agency usually guides the intern through the process. Most interns receive a Student Visa OR Cultural Exchange Visa depending on their program. For more information on visa, consult the section on Passport and visas.