Here’s a recipe that works on Ubuntu & friends:

1. Unzip the WordPress files.
2. Prepare a database in PhpMyAdmin. The collation should be utf8.
3. Edit /etc/apache2/envvars (see below)

How to edit the  /etc/apache2/envvars file

locate lines:

export APACHE_RUN_USER = www-data
export APACHE_RUN_GROUP= www-data

Change this to:

export APACHE_RUN_USER = yourName
export APACHE_RUN_GROUP= yourName

The username should be someone who can read, write and execute the files.