The problem could be with your; cable, phone or computer.
I'd try a different cable you might have a dead cable. Also if you aren't using the supplied cable you can have problems some cables are not fully wired so only work for power and not for data, I have a cable like this that came with a USB power adapter, it works fine for charging but that's it.
Have you got a different PC you could try the phone with?
Does the phone say anything when you connect it to the PC?