Open Apache2 config file:

sudo nano /etc/apache2/apache2.conf

Add these lines in the end, change your_username to yours:

# UserDir #
UserDir public_html
UserDir disabled
UserDir enabled *your_username*
Options Indexes FollowSymLinks -MultiViews AllowOverride None Order allow,deny Allow from all

Make symbolic links for mod_userdir to enable it:

cd /etc/apache2/mods-available/
sudo ln -s ../mods-available/userdir.conf
sudo ln -s ../mods-available/userdir.load

Or enable the module without editing any files:

a2enmod userdir

Remember to restart Apache:

service apache2 restart