Visa for Internship and Travelling

I'm planning to travel to China from the end of June to the end of August, and during that time I'll first be visiting relatives in Hangzhou and taking a trip to Japan, and then complete an internship at Beijing Normal University, after which I want to travel some more in PRC. What type of entry permit should I apply for because my internship is only 3 weeks long and Beijing Normal University says that their letter of admission will only cover that duration? I guess I would want a double entry, but would I need 2 separate visas to do tourism and work or just one?
(I am a British citizen but most of my family are in PRC.)