Is it normal for a 2011 MBP to display "Not Charging" while using Skype?
When you have Skype running, and you see either slow charging or fast battery draining (depending on whether or not it's plugged in), launch an application called Activity Monitor. You can launch it from Spotlight, or from the Finder when you go to the /Applications/Utilities folder, or from Launchpad if you go into the Utilities folder.

With Activity Monitor open, click on the column header that says "% CPU", and click it again if needed to make the little triangle point down. This will show your running applications, with the ones using the most CPU at the top. Also, you can select "CPU" from the tab bar near the bottom of the window. There you'll see a graph of the load (or demand) on the CPU.

This will give you indications of whether the CPU is working excessively, and which applications are causing the issue. It's a fair assumption that Skype is using a lot of CPU for some reason. Do some observations, then go to the Skype support webpage (for US English, go here: https://support.skype.com/en-us/).

If you find that some other process or application is causing the high CPU usage, leave a comment here or update your original question with your findings.