SolidWorks is not the proper tool for this.
That being said, the only easy option I can suggest is zsurf:
For an example of zsurf look at this thread: https://forum.solidworks.com/message/234998#234998
Otherwise, you need a mesh/sub-d modeler. Those kinds of modelers are much more adept at this type of modeling, and are really what you want. modo, 3Ds Max, Maya, Lightwave, Bryce, Form-Z, Zbrush, etc. Blender is an option that is free. But this is not, I'm sure, what you were looking for by using your existing knowledge of and access to SolidWorks.