No you cannot get a working viza (Z) once inside of China, you will have to leave the country to do this. To get a Z Visa you will need a letter of invitaion from your future employer and you will also need a government document which agrees with this. If you have just the letter of invitation you can get a F Visa (business), but will have to leave to have it changed to Z.

You can work illegally with an L Visa (tourist) and take your chances of abuse. Head into Yangshou in Guanxi province. There are a lot of L Visa jobs there. And it's a spot for backpackers.