Updates
Step 1
To install a new version just replace the entire directory with the new package EXCEPT the file autoconf.php which can be found inside the /includes folder.
Additionally you have to set write permissions to the temporary/ folder.
Attention: Kimai will NOT work with wrong permissions!
For Linux that could be:
chmod -R 777 temporary/
It would be even better, if you give writing permission to your PHP or Webserver user only, instead of making it world writable. The last can be or is a risk, especially in shared hosting environments.
Step 2
Visit your Kimai installation with your web browser and the auto update process will perform all necessary tasks.
A new version updates the database automatically. Before that a backup of your data will be created. Normally you'll not even notice this process. If anything goes wrong during the database-update you can recover the data via phpMyAdmin.