Purchasing a used C300 tomorrow most likely.
What should I be looking for and what tests should I do before purchasing it to make sure everything is legit and working well?
Check the XLR and monitor cables are working.
Inspect the sensor.
Make sure the ND wheel is working as well as all the other buttons.
Check the white balance is functioning properly.
Roll off some footage and have a good look at it.
If the camera has been sitting unused for several weeks, make sure it's holding internal memory settings. The internal battery can discharge. If a problem, plug it in to AC for 24 hours, powered up. If that doesn't recharge the battery, the camera has to be sent into Canon.
Make sure it's a US/Canada unit. Couldn't hurt to contact Canon Cinema EOS support to double check that the serial number is in the system, that it's updates performed, like dust seal around sensor/ND assembly and any repair history. I once almost bought a used, expensive 2/3" camera, until the manufacturer told me that it's repair history showed that it had been sent in for water damage and then repairs were declined. If camera under warranty, check to see if it can be transferred.
Shoot a white card at f16-22 to see any dust on the sensor after blowing dust off of the filter in front of the ND's and cleaning the lens.
I'm sure things will check out, but can't hurt to be very careful. Congratulations! The C300 is the most fun to shoot with of the dozens of cameras I've owned over the past 30 years!
Congratulations! Some good advice already given. If there is dust in the sensor, then using f16-f22 would show it up in the LCD monitor in shooting mode.
The downside, one producer didn't believe that this is a cinema camera because of its small size.