Reliably? You can't, yet.
You can catch the majority of changes by firing your update checker onkeyup as well as onchange. However whilst this picks up simple keyboard interactions, it fails to trigger on other events like cut-and-paste (there is onpaste in some browsers, but you can't rely on it), drag-and-drop, undo-redo and spellchecker changes.
HTML5 proposes oninput to solve this, but the implementation isn't there yet. If you really want to spot all input changes today you will have to setInterval a function that keeps checking the input's value to see if it has changed.