If you don't expect too many visitors on your web-site, you may install free stream broadcasting server software like red5. You may also use commercial server software like Flash Media Services,... Just make sure that your web-server(s) doesn't experience any overload and that your hosting provider allows to use as much traffic and server load as you need or more.

When you have a really large audience instead of your own servers you may use a Content delivery network (CDN).

There are lots of scripts of YouTube clones that you may use. If you need a custom web-site to be developed, you can always outsource your project via a freelance marketplace. They allows you to hire professionals from the U.S., Canada, India, Russia,.... And you can always select a person or a web-design company that possesses the skills you need. When choosing a freelancer you can rely on the feedback of other buyers of the marketplace. You should also evaluate his/her resume and/or portfolio.