If you need to open a VPN connections to Softlayers private network, you should do this via a browser, as described at http://knowledgelayer.softlayer.com/procedure/ssl-vpn-connections. From what I experienced, with Linux you would need to run a browser as root to follow the instructions.

Fortunately, there is another way: The command-line client. Using that is rather straight forward (except one gotcha).
1.        Download the VPN software from http://speedtest.dal05.softlayer.com/array/ArrayNetworksL3VPN_LINUX.zip
cd ~/Downloads
wget http://speedtest.dal05.softlayer.com/array/ArrayNetworksL3VPN_LINUX.zip
2.        Rename the zip file to bin (I tried to uncompress it, which failed miserably)
mv ArrayNetworksL3VPN_LINUX.zip ArrayNetworksL3VPN_LINUX.bin
3.        Make the file executable
chmod 777 ArrayNetworksL3VPN_LINUX.bin
4.        Execute it as root
sudo ./ArrayNetworksL3VPN_LINUX.bin

And your are done.

Now you can open up a VPN connections to Softlayer from the comfort of your command-line:
sudo /usr/local/array_vpn/array_vpnc -hostname https://vpn.ams01.softlayer.com -username [YOUR SOFTLAYER USERNAME] -passwd '[SUPER_SECRET PASSWORD]' &

Which should give you a

array_vpnc: VPN TUNNEL SUCCESSFUL!

message back.

A quick "ifconfig tun[x]" should also show a valid 10.a.b.c Softlayer IP adresss.

ifconfig tun1 
 tun1      Link encap:UNSPEC  HWaddr 00-00-00-00-00-00-00-00-00-00-00-00-00-00-00-00   
          inet addr:10.1.16.244  P-t-P:1.1.1.1  Mask:255.255.255.255 
          UP POINTOPOINT RUNNING NOARP MULTICAST  MTU:1450  Metric:1 
          RX packets:1 errors:0 dropped:0 overruns:0 frame:0 
          TX packets:0 errors:0 dropped:0 overruns:0 carrier:0 
          collisions:0 txqueuelen:500 
          RX bytes:20 (20.0 B)  TX bytes:0 (0.0 B) 
 

If you can also successfully ping 10.0.80.11 or your server's private address,  you are successfully connected.

ping -c 3 10.0.80.11 
 PING 10.0.80.11 (10.0.80.11) 56(84) bytes of data. 
 64 bytes from 10.0.80.11: icmp_seq=1 ttl=253 time=35t4.6 ms 
 64 bytes from 10.0.80.11: icmp_seq=2 ttl=253 time=34.9 ms 
 64 bytes from 10.0.80.11: icmp_seq=3 ttl=253 time=35.1 ms 
 
 --- 10.0.80.11 ping statistics --- 
 3 packets transmitted, 3 received, 0% packet loss, time 2003ms 
 rtt min/avg/max/mdev = 34.605/34.886/35.148/0.222 ms

Pro-tip: Use the VPN server of the Softlayer datacenter where your machines reside.  Apparently, there exists at least one Softlayer VPN Server for every datacenter:

• https://vpn.dal01.softlayer.com for Dallas, Texas, USA
• https://vpn.sea01.softlayer.com for Seattle, Washington, USA
• https://vpn.wdc01.softlayer.com for Washington, D.C., USA
• https://vpn.ams01.softlayer.com for Amsterdam, The Netherlands, Europe
• and so on ...

I must say, that I was pleasantly surprised (again) by Softlayers support. I had a correct answer to my question in under 5 minutes after opening the ticket.