OpenVPN - Linux client connects but could not access internet, routing issue

I could solve the issue by lowering the "metrics" of my ethernet/wifi card. you can lower the metrics (priority) of the routing entries for your network card via the "route" command or by using the additional tool "ifmetric" (sudo apt-get install ifmetric) i.e. ifmetric eth0 100 (0 = highest priority)

After that I recognized that the DNS entries from the openvpn server has not been taken over when vpn connection has been setup. so I googled around and I found an answer that you need to add on the ubuntu client the following lines on its config file:

script-security 2
up /etc/openvpn/update-resolv-conf
down /etc/openvpn/update-resolv-conf

This updates the DNS entris in the "/etc/resolv.conf" when the vpn connection is established by using the pushed entries from the server.

So it works now as a charm.