You could use JavaScript and trigger the hidden file input when the button input has been clicked.
http://jsfiddle.net/gregorypratt/dhyzV/ - simple
http://jsfiddle.net/gregorypratt/dhyzV/1/ - fancier with a little JQuery
Or, you could style a div directly over the file input and set pointer-events
in CSS to none to allow the click events to pass through to the file input that is "behind" the fancy div. This only works in certain browsers though; http://caniuse.com/pointer-events