The best, most professional way to handle this is to bring your concerns confidentially to your own manager or HR and let them handle it. I used to work in HR addressing issues like this and taking things into your own hands to gather "proof" is definitely not appropriate and, at the very least, calls into question your own work ethic (how much time did YOU take away from your own work to capture this video?) and professionalism (it's not YOUR job to address the issue and there's absolutely no reason to go that far to provide proof of the problem).
Just schedule a meeting with your boss and describe your concerns specifically, e.g., "I noticed Bob has been sleeping at his desk lately and it's impacting the team and the work we do in X, Y, and Z manner...."