There are plenty of applications to do that, such as
http://www.phpmotion.com/ however, coding a service like that takes time, and you should probably host your videos on AWS or something similar, therefore, if you end choosing this way it's in my opinion going to cost you more.
For example, you hire a developer to code it for you, he expects you to pay him $10/hourly rate, and a good looking - working- video sharing site isn't going to be coded in less than 10 hours, then you've to setup the application and blablabla.