There's nothing built into DW that will do this for you.
But, if you're hosting your video yourself, you could set up Events in Google Analytics to track pretty much every aspect from clicks to % of the video played. GA would also show you all of the other info it tracks when compared against the video Events you set up.
It won't show it on your page (your visitors are unlikely to care about a hit counter in the least), but it would tell you or your client everything you'd ever want to know about the people clicking your videos.
The first step would be setting up an Analytics account, then you'd just need to set up the Events you want to track. All of the info is available through the GA site.
I tried but I am hosting on a server at work which is only available to our network users. Getting reports is a long process here so I was trying to find a way around where I can do my own count/views on the videos.
How much access do you have, to what kind of the server?
A view tracker is going to require the ability to use some kind of scripting to write to a file on the server every time the video is viewed. There are simple hit counter PHP scripts all over the internet that you could use, if you have PHP installed.
Another option, for generating full analytics style information on your intranet site, without having to rely on the the IT guys to generate reports, would be to have them install PIWIK on the server https://piwik.org/
That would give you essentially a Google Analytics style dashboard to log into that tracks usage locally.
You could do this with PHP and flat text files (no databases required).
See link below for a free package that includes a backend Admin area for tracking clicks/downloads.