I applied through college or university. The process took 2 months. I interviewed at Arizona State University (Tempe, AZ) in Feb 2014
Interview
For simple process. You get accepted to the university by submitting relevant application material (i.e. GRE, PGRE, Personal Statement) and do well in classes. In essence, the interview process is an assessment of your capability.
Other Teaching Assistant Interview Reviews for Arizona State University
Interviewed as a TA. It was a quick meeting with a professor. I had to demonstrate my skill related to the field and I had to demonstrate my teaching skill.
I applied through an employee referral. I interviewed at Arizona State University
Interview
Call from professor asking about my experience in the field and how confident I am to answer students questions and help them in labs. Not too hard, good course and good professor, and good university.
It is easier to get through if you make an acquaintance with the professor teaching a particular course of interest. The professor can recommend you to the hiring department and make the process much more simple. I did not have an HR interview as a result but just interaction with the professor