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Sharing video on shared hosting is not a problem if you have the correct script. Where you may run into issues is if you are looking to encode videos on shared hosting. In that case you want FFMPEG hosting.
I've offered shared hosting in the past and we didn't allow people to upload videos that they had to decode and whatnot with ffmpeg. If it's just a simple video that only requires bandwidth to stream it then I don't see a problem.
Some time ago, I allowed video hosting and it was highly abused. It resulted in a lot of sever resources being hogged at certain times of the day and it just turned out to be a mess. I wouldn't recommend offering it. There are other video hosting solutions available and it's not considered unprofessional to link to videos on sites like YouTube or DailyMotion.
I'm not sure how different cloudlinux is, nor have I ever used it.
On my box, I just made sure the rpmforge repo was installed, and then used yum. If I remember right, I think you also have to install the necessary codecs... it's been quite some time since I installed it.