How do you package a computer processor?

It is actually fine in the plastic CPU tray you have it in. I was expecting a pic without one, and was going to mention a CPU tray. But you have one. So, now encase the whole thing (CPU/tray) in bubble-wrap, and tape it up with scotch tape. Then simply mail it off in a padded manila envelope from the post office or the UPS/Fed-EX station. That is how I've mailed a couple CPUs off to people in the past. No issues at all. Provide tracking, of course.