I found out about Koch's internship program through my university's career fair. I filled out the application on the Koch hiring website. Filling out the application is easy and quick: you upload your resume and it is parsed for information, so a lot of boxes are automatically filled out with what you put on your resume. There is also a short section of behavioral questions. About a week later, I was informed that I could schedule a time for an on-campus interview. That interview was about half an hour long and fairly easy. All the questions were about my behavior. The interviewer did ask me a little bit about my technical knowledge to find out what sort of position I was looking for, but the interviewer emphasized that Koch does not expect interns to have extensive technical knowledge.
A few weeks after that first interview, I was informed that I had made it to the second (and final) round of interviews, which take place in Wichita. Koch paid for a hotel room and reimbursed me for travel expenses. This final round of interviews consisted of two back-to-back hour-long interviews. The first interview was, again, focused on behavioral questions, especially on dealing with difficult customers/co-workers and process improvements (what processes I use to solve problems). The second interview was more technical in the sense that I was asked what programming languages I had experience with and what sort of programs I had created. This interview was still mostly behavioral however, and was focused on the same kinds of questions. The interviewer emphasized that Koch focuses on finding interns who fit the company values, rather than just focusing on technical skills.
Read up on the Koch guiding principles and look up common behavioral questions and you will be fine. And, of course, be prepared to answer basic interview questions ("Where do you see yourself in 5 years?" "Why do you want to work for Koch?" "Why software development?" etc). Everyone I talked to was laid back and personable, which helped the process go smoothly.