My tourist visa runs out on the 2nd July but I need to stay in China until the 15th,..

It is possible to extend within Beijing. Fee is RMB 168. Following documents are required:
1. Passport with current L
2. Registration Form of Temporary Residence in Beijing
3. Visa or Residence Permit Application Form
4. A recent 2-inch photo with white or light blue background
5. Documents certifying financial support for living cost in Beijing (e.g.certificate of deposit, at an average of US$100 per day).

Thanks,
do you know where the best place to do this is, in Beijing? I am based in Dongzhimen

You need to do this at Beijing Municipal Public Security Bureau
Address: No.9, Dongdajie, Qianmen, Dongcheng District, Beijing