How do I use an Apple Gift Card on my Apple TV to rent movies?

Here is what I learned.

1. Receive an email from Apple with a Redeem link.
2. On the web page I had the option to Redeem my Apple gift card at an Apple Store (hardware) or on iTunes or the App Store (software).
3. I choose to redeem my Apple gift card using iTunes or the App Store.
4. Using iTunes, I selected Account, View My Account and then Manage Payments where I saw the message, "Your Apple ID balance will be used for each purchase first until it reaches zero" and I saw a credit/gift card icon for my gift card and the credit card I had on file.
5. Using Apple TV, I selected Movies iTunes, navigated to a movie I wished to stream, selected the Rent option, was prompted to enter my security code for my credit card and then was able to watch the movie.
6. Getting out of the Manage Payments window in iTunes and then selecting to get back into it, I could see that the balance of my gift card Apple ID had changed from the original gift card amount to an amount less the cost of the movie rental.
7. I submitted Apple Feedback to encourage Apple to display this same message on the Apple TV software when customers are selecting to rent or buy movies or TV programs so that they do not think that their credit card is being charged and I suggested that they not require customers to enter their credit card security code if they have enough Apple ID balance from a gift card to make the payment. In this way customers will know what their Apple TV is really doing in relation to their account.

Technically, your answer is correct in that I could not redeem my gift card directly using Apple TV but I could redeem it to be used in iTunes or the App Store and so since my Apple TV allows me to rent iTunes Movies or TV programs, this transaction will be paid for by my gift card and the answer to my question is yes, I can use an Apple Gift Card to rent/buy movies or TV programming on my Apple TV. It is just not that obvious that I can do that or that the Apple TV transaction would work in the way I intended, namely being paid for using my gift card. There are easier more intuitive ways to program all of this that other companies use by displaying icons of gift cards and credit cards on file and allowing customers to select which they will use for payment. Apple uses Continuous Process Improvement (CPI) techniques in the design and development of their hardware and software with rigorous testing and so I suspect in the not too distant future they will improve all of this so that it is more intuitive and self evident to customers who use Apple TVs how they can make sure their gift card is being used for rentals and purchases until it is used up.