Using your Chromecast while travelling in Hotels won’t be as simple as connecting to the TV and setting it up with the Hotel provided wireless. First issue is that the Chromecast itself probably won’t get internet access due to the captive portal security (accept disclaimer) for internet access. The second issue is that intra-SSID traffic is likely blocked so your phone/tablet won’t be able to communicate with the Chromecast while both connected to Hotel wifi.
The solution for the above issues is to connect a travel wireless AP/router to the Hotel room wired connection and broadcast your own SSID which will be shared among all your wireless devices. My favourite travel router is the TP-Link TL-WR703N. The $25 unit is a very compact AP that can be powered via USB on your laptop (it also comes with the power adapter). I’ve purchased of few myself from both E-bay and DealExtreme.
The next best travel router is the TP-Link TL-WR702N which is widely available in North America, costs a tad bit less at ~$20 and comes with english firmware. It lacks the USB port for a 3G connection and the ability to flash custom firmware such as OpenWRT or DD-WRT.