Use HTML5 video. You can search the forum for it but you can start with one oof my post : Review of EasyHTML5Video generator app: it's very good! - Webflow Tips - Forum | Webflow
Now know about the downsides : it doesn’t work on mobile… on mobile it will show a play button under the overlay layer.
So the solution I use is I completely deactivate the video bg element for mobile and instead switch to an element sporting an animated GIF version of the video (shorter). It works great. You can see that in action here: http://www.1001pneus.com/
Hosting: You can use any server you can access with a FTP, or simpler, use Dropbox. A good practive is to create a /Cloud folder in your dropbox where you’ll put all the assets you need to access for your project. Then one subflder for each project for you to remember what to keep, what to clean. Then put your video and jpg (poster) files in it. Right click and “Copy sharing link”. Then, important, replace www.dropbox by dl.dropboxusercontent in the url you get from dropbox, and test each url in a browser to be sure it really points directly to each video file.
I hope this helps 