You can aquire a visa for China in Saudi. If your visit is for sightseeing, you'd better contact a travel agency to join a tour group. The embassy's site states that individual tourist application is not acceptable.

The person I plan to visit in China is a Chinese citizen. She lives in Chong Qing. Is there a way I can obtain a Visa to visit her? We plan to travel to Bejing for three days, but will stay in Chong Qing the rest of the time while in China.

Mr.Cloward, yes, you can get a L visa to visit her. You should provide the following requirments for L visa application:
1/ Original passport with validity more than six months and minimum 2 blank visa pages;
2/ Photocopy of the first 2 pages (personal information) of the passport and the recent Chinese visa page if there is any;
3/ One fully completed visa application form in Chinese or English, attached with one of the applicant's recent color photograph (30mm X 40mm);
4/ Due certificate or document supporting your relationship with the person in China you are to visit (original and photocopy);
5/ Invitation letter;
6/ ID card copy or passport copy of your person you are to visit;
7/ Return flight tickets and their copies.

You should apply for chinese L visa at Chinese Embassy in Riyadh.