If you buy Skyrim for PC and play it and then buy a new PC a few months later. Is there a way when you install it on your new PC to transfer your character or would you have to start from scratch?

Yes you can transfer your character by just saving the save files and putting them in the corresponding folder on the new computer.

Skyrim is tied to your Steam account, so aslong as you have a save on the steam cloud you can play on another pc.