Extending classes can't be done using built-in means. You'll have to write some code to make it work.
I've put together a utility library last week that takes care of most of it, it's on Github: https://github.com/SamVanheer/AngelscriptUtils
The code you want is the extension support, which is defined primarily in 3 files:
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/Angelscript/util/CASExtendAdapter.h
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/Angelscript/util/ASExtendAdapter.h
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/Angelscript/util/ASExtendAdapter.cpp
CASExtendAdapter is a template class that is inherited from to implement the adapter between the C++ and script class.
The other files define helper functions to define script code that scripts can inherit from to extend from the classes.
For an example of its use, see these classes:
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/test/CBaseEntity.h
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/test/CScriptBaseEntity.h
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/test/ASCBaseEntity.h
The typedefs are critical for the macros to work. Otherwise you'll have to write a lot more code to handle each call separately.
The script API is just as important as the C++ code:
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/test/Main.cpp#L112
https://github.com/SamVanheer/AngelscriptUtils/blob/9c024dfcf80ff280cd96f4ffa714de0efb291e0e/src/test/Main.cpp#L184
Registering the C++ class API, the base class method call type and the interface you can use to take handles to script extensions is an important part.
Creating the instance is handled like this:
https://github.com/SamVanheer/AngelscriptUtils/blob/master/src/test/Main.cpp#L337
The library is a WIP, but this should let you get started with your own code.
Based on your script code and what you said about it not being set up correctly, you're still having problems with getting Angelscript integration done. I suggest getting that set up before trying to integrate class extension support.