You should be able to install a Laravel app just like any other website. See How To Set Up Apache Virtual Hosts on Ubuntu 14.04 LTS.

You'll need to install a couple of PHP extensions that Laravel needs:

sudo apt-get install php5-mcrypt php5-json
sudo php5enmod mcrypt
sudo php5enmod json
sudo service apache2 restart

The rewrite module should be enabled by default, but if your routes aren't working properly, try enabling it:

sudo a2enmod rewrite
sudo service apache2 restart

Finally, you need to grant Laravel write permissions to the app/storage directory:

sudo chown -R www-data:www-data /var/www/domain1.com/app/storage

For users of the Ubuntu LAMP 1-Click droplets, to this I would also add something I figured out the hard way:
Often Laravel setups will have the app located in something like /var/www/laravelapp with the public folder at /var/www/laravelapp/public
The Laravel documentation points out the need to edit your apache2 sites-enabled configuration to change the DocumentRoot to /var/www/laravelapp/public

You must also change the <Directory> for the .htaccess rules to match! Otherwise even after you enabled mod_rewrite, your .htaccess in /var/www/laravelapp/public will be ignored, and nothing but your home page will load.