Your question is far too broad, you're essentially asking how to build a website. I suggest you get a book on MVC and go through it, or at least go through some tutorials online, and how to do these things will become clear.
To answer your specific question the easiest way is to use jQuery and the click event;
.click() | jQuery API Documentation
or more correctly the change event
.change() | jQuery API Documentation
However that won't get you far in your quest, you'll also need to know how to use ajax to call controller methods, how to pass the data on the page, how to search the database, how to show the results etc etc. Handling the click event of a checkbox is a very small part of the functionality you want to implement.