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I have a few recommendations. First on your computer, verify you can sign into the iTunes store without issue.
If so, then I would take a look at this article for the recommended wifi settings for your device:
iOS and OS X: Recommended settings for Wi-Fi routers and access points
Next if needed, I would make sure this port number is open on your router:
TCP port 80 and 443 are used for basic and secure communications with the iTunes Store via the Internet.
From: Apple TV: TCP and UDP ports and protocols used