I applied online, and then talked to a recruiter at a career fair. About two weeks later, I got a phone call from an HR manager, who set me up with a phone interview with a power electronics engineer. In that interview, I was asked technical questions relating to power electronics, control systems, EMI testing, and especially the engineering experiences I listed in my resume. There were no "personality" questions, except "what does SpaceX do" and "why do you want to work at SpaceX". I was only able to answer about half the questions.
I was contacted the day after the interview for a second phone interview, with a different power electronics person. This time, the questions were very in depth about the specifics of my projects. How exactly my constant current regulator worked, etc. This time, I was able to answer very well. He also asked if I had any PCB layout experience - which I have.
I received my offer soon after the second interview.