How do I go about getting a refund for an apple tv show purchase that was done accidentally?

Apple's policy on sales of digital content is that all sales are final. If however you wish to appeal to Apple, select the content which is causing a problem and use the 'Report a problem' button in Your Purchase History on your computer.

Currently, if your purchase happens to be your most recent purchase you will not be redirected to the report form, in this case use the report a problem option from the invoice that was emailed to you.