The definitions you give to EventHandler are only active when your selection is on its contents. An example of where the button press also works would be:
EventHandler[
InputField["Click"], {{"KeyDown", "c"} :> MessageDialog["This is button"],
"MouseClicked" :> MessageDialog["This is button"]},
PassEventsDown -> True]
Where you will get the pop-up first when you press the field, and afterwards if you press c while the selection is in the input field. Notice I added PassEventsDown otherwise your cursor will not go into the field when you press it the first time.
If you want to have a global event that triggers when you press c, independent of where the selection is, you can add it to the NotebookEventActions of the notebook:
SetOptions[EvaluationNotebook[],
NotebookEventActions :> {{"KeyDown", "c"} :> MessageDialog["This is button"]}]
Then any press of c will trigger the event. And if you need to get back (because it gets annoying having a pop-up any time you press c for instance) you simply set the NotebookEventActions back to {}
SetOptions[EvaluationNotebook[], NotebookEventActions :> {}]