Hotels that give you free wifi still make you click through an "I promise not to be an idiot" form button before you can go online. Your devices (chromecast, Apple TV, Roku and other boxes) for the most part don't have a browser to click that button. That's the issue.
The basic concept is that your creating your own wifi network just like at home. Think of the Ethernet port in your hotel room like its your modem at home.
1 Hook up your router (power, Ethernet/cat5 cord). Your router should be set to AP mode that makes its own network. Not joining a network.
2 Connect to your router's wifi on your phone, ipad, laptop and enter your password (if its your first time). Don't power-on your ATV/chromecast yet.
3 Open a browser and try to go to any page. The hotels wifi will hijack you to their page. Click the OK button (whatever their version of OK is)
4 Plug in your Apple TV /chromecast and login to your wifi
5 Thats it
If you were expecting a bunch of smoke and mirrors with secret handshakes and knocks, so sorry to disappoint. The hotel's system has agreed that your a swell guy because you clicked OK. That gave your router's MAC Address a pass. It just so happens that MAC Address is a router that shares that connection with your other stuff. Once in a while you may need to do this process again because they like you to click that OK every now and then. If you notice you can't get your Apple TV online you'll know its time to restart the process. I like to pull the plug on the Apple TV during this process so it's not trying to get internet access while your clicking the OK button. Same for the chromecast. If you can't connect just pull the plug on the router and your other stuff.
-- This can also be done with a windows application called Connectify or by using a wifi repeater if there isn't an Ethernet jack in the hotel room- Maybe I'll write up a tutorial on doing this with a wifi repeater at a later date, but I haven't seen a hotel room without Ethernet jack near the desk since the 90's.
This 'router way' of doing this is a lot easier than the Connectify or router repeater way.
I'd also like to point out that it might be easier to hook up your phone, tablet, laptop directly to the tv. My issue is I hate tying up my phone and not being able to use it. Hotels are also making it harder to get to the hdmi input. Google the tv model for the manual if you need to.