You really shouldn't be sending an http status of 500 on an expected registration failure such as "email already exists" condition... this should be handled by a parameter that denotes success / failure as part of a 200 response.
You can handle unexpected internal server errors (status 500) using success or error callbacks like so:
$.ajax({
url : "../api/Register",
type : "post",
data : {"Email" : "you@example.com", "Password" : "pw"},
dataType : "json",
success : function(response){
// handle http 200 responses
if(response.registrationApproved){
alert("Thank you for registering!");
}else{
alert("email exists");
}
},
error : function(){
// handle 500 or 404 responses
alert("server call failed");
},
complete : function(){
// if needed.. this will be called on both success and error http responses
}
});