In general, this is a DNS matter and it is a little bit tricky.
You need a PC to connect to the hotel LAN port directly and launch any webpage. When you are prompted with the hotel's login page, don't login and check the followings :
1) The URL of the hotel's login webpage (e.g. www.yourhotel.com/login.html) from the browser
2) At DOS prompt, run IPconfig and check the DNS server's IP address of your hotel network (e.g. X.X.X.X) at the Ethernet Adapter section.
Then disconnect your PC from your hotel LAN port and connect your WIFI router to the hotel's LAN port
1) Change your PC's WIFI configuration, primary DNS -188.8.131.52, secondary DNS-DNS server's IP address of your hotel network
2) Connect your PC to WIFI router using the wireless card, after connection go to the WIFI router's admin page and ensure your router can get the proper IP address from the hotel's network.
3) Use browser to launch the hotel's login webpage (e.g. www.yourhotel.com/login.html) and you should get the login screen. Click the appropriate button to login and you should be able to get to internet afterwards.
However, you should first check if your router can get local IP from your hotel's network. With the router's LAN port connected to hotel's network, you should access the router's admin page to check via WIFI. You should be able to get the local IP from your hotel's network without the hotel's login screen. If your router cannot get Local IP from your hotel's network, there is no way you can proceed further.