This is mostly interesting, but one angle is to create a simulated key event:
function simulateKeyEvent() {
  var event = document.createEvent("KeyboardEvent");
  event.initKeyEvent(                                                                                      
             "keyup",
              true /*let event bubble up*/,                                                      
              true /*can the event be canceled?*/, 
              null, 
              false /*ctrl key held down?*/,
              false /*alt?*/,
              false /*shift?*/,
              false /*meta (cmd)?*/,
               9 /*keyCode*/,                                                      
               0);
  var cb = document.getElementById("checkbox"); 
  var canceled = !cb.dispatchEvent(event);
  if(canceled) {
    // A handler called preventDefault
    alert("canceled");
  } else {
    // None of the handlers called preventDefault
    alert("not canceled");
  }
}
Now, to actually solve your problem, you might want to run the code outside the extension isolated world, in the page's javascript. To do this, I have a nifty function for you:
function runInPage(code) {
  var script = document.createElement('script');
  script.innerHTML = code;
  document.documentElement.insertBefore(script);
}
Then try runInPage('$(".userItems input.amount").keyup();');
But you'll probably need to load jQuery before this, so do:
var jq = document.createElement('script');
jq.src = 'http://ajax.googleapis.com/ajax/libs/jquery/1.7.1/jquery.min.js';
document.documentElement.appendChild(jq);