Dvd drives and HDDs are the most unreliable components for a laptop - that is they quite working more often than other components.
Probably the simplest way is to test it on a Desktop or a diff laptop. Cheap and easy as long as the connector on the back of it is the same as a 2 1/2 inch Laptop HDD.
.. Removing should be just removing one screw (I've replaced the DVD drive on two of my toshibias and replaced with a blu-ray Rom/DVDRW drive.
.. Buy a SATA -> USB cable adaptor (about 12 bucks) (PS They are handy for use with laptops anyway).
http://www.newegg.com/Product/Product.aspx?Item=N82E16812123312
Added I prefer, for 6 Bucks more, this ( it comes with software to clone a Laptop HDD to a New HDD or a SSD):
http://www.newegg.com/Product/Product.aspx?Item=N82E16812161004
NOTE with this cable should plug it into a USB Powered Hub, not directly into the USB port.
If it does not work - it's the drive, if it works it windows/driver/software problem.