Assuming you’ve already installed apache, do the following:
sudo a2enmod userdir
sudo service apache2 reload
The first command enables the userdir apache mod, which does exactly what you want. The second reloads apache configurations so that it starts using the new configuration.
To install apache2:
sudo apt-get install apache2
Of course, you’ll also need to make sure that the permissions on your public_html folder allow the www-data user to see the files in there — 755 usually works well. To do this:
mkdir ~/public_html
chmod -R 755 ~/public_html
This will recursively (-R) go through your public_html and set the permissions to 755 (owner rwx, and both group and other r-x, r-x).