I was staring at the window one day. My advisor came into the lab, and offered me the job. I said I would take it. That was the whole process... But realistically, if a person who is a grad student at the Mechanical Engineering department wants to become a TA, they will have to talk with the head professor who is in charge of the entire graduate program. Let the head professor know that you are interested in becoming a TA for the coming semester. He will keep your profile on file, and let you know before the next school year begins. It's more like an evaluation of how well you are doing at school.