Is It Possible to Make Caucasian Eyes Look Asian? (photo)

Attempting surgery to achieve this would be a mistake.  Especially to achieve a "Korean" appearance, your eyes could easily be made to look more Korean with makeup.  The size and shape of your eyes and supratarsal crease (double eyelid) are not beyond the realm of what some Korean girls achieve surgically.    

While you do have Asian ancestry, that helps with the overall look of your face. In our practice, we do procedures to enhance the shape of the eyes ("exotic eyes" or almond-shaped eyes).  Creating an Asian epicanthal fold and making the eyes more puffy are atypical procedures (and one which I would advise against as the results may not look natural). The use of makeup can sometimes help to achieve this look. It's important for you to meet with a specialist who can guide you with the types of procedures that realistically will benefit you. 

I suspect that what is missing to make you feels are more asian in appearance is an epicanthal fold so that your outside fold is and inside fold.  This is simply never going to happen.  I would be very cautious in searching for surgery on your eyelids.  Eventually someone will offer you surgery whether it is right or wrong.  Sometimes the very best surgery you get is the one you don't have.