When any user clicks on "Responds to Survey" button/link I want to perform some operation using code behind but can't figure out how can I do it or which event handler can be used.
You cannot use any event handler as its not a button but just only link to Newform.aspx for adding new responses to survey list. you can create a custom list with custom newform.aspx form with page_load method to do the code behind you want to do.
surveys are lists and with event reciver you can only do the following as the are lists:
Item adding - Item added
Item updating - Item updated
Item deleting - Item deleted
for more info look here:
add a hidden feature that is activated on that page. in codebehind of feature you find the control and create an event handler and do what you want in that click event.
hope it helps