The other answers are on the right track with mod_userdir, but using that will give your website the base URL http://www.yourdomain.com/~username/ - for instance, a file /home/username/public_html/index.html would be accessible as http://www.yourdomain.com/~username/index.html. If you want your files to be accessible under the domain root, as http://www.yourdomain.com/index.html for example, then you'll need to put the directive
DocumentRoot /home/username/public_htmlin the Apache configuration file.