Download Glassfish application server (in my case promoted build http://dlc.sun.com.edgesuite.net/glassfish/4.0.1/promoted/).
Unzip glassfish-4.0.1-b01.zip and start server
Download Activiti (here http://www.activiti.org/download.html)
Deploy Activiti war-files to Glassfish by using the admin ui (http://localhost:4848)
Start Acitiviti Explorer (localhost:8080/activiti-explorer)
The deployment of Activiti 5.13 on Glassfish v. 4.0.1 works perfect out of the box!