Try this process:
1. Start a new part.
2. Start a sketch on a plane.
3. Through pull down menus, tools>sketch tools>sketch picture
This should bring your picture into your skech, then you will have to trace your picture with the normal SW sketch tools. If a .ai file doesn't work, convert your .ai file into a jpeg, or another file type that SW will work with.
You may also be able to export the Illustrator file into DXF or such, too, and import this into SolidWorks. However, this often imports as a million tiny line segments, in which case it won't work very well for cut paths. Might be worth trying, since tracing takes a while (if you're accurate), perhaps with different settings on import/export to see if you find the right flavor.
I have done this in the past, but don't have Illustrator installed on my new machine.
Have you tried dragging and dropping the .ai file directly into an open SW part, or into an open sketch in a SW part?