Chromecast (1st generation) is only compatible with 2.4 GHz Wi-Fi 802.11 b/g/n Wi-Fi networks, so it won't list 5 GHz networks when setting up the device. Please make sure the Chromecast is using the 2.4 GHz band. Chromecast (2nd generation) is compatible with both 2.4 GHz WiFi 802.11 b/g/n WiFi networks and 5 GHz networks, so you will see both when setting up the device. If you have a dual band router, you must disable AP isolation. Other devices such as your laptop/computer can use the 5 GHz band; the 5 GHz band will maximize performance when using the Cast extension to cast a tab.
If you have enabled AP Isolation/Client Isolation on your Wi-Fi router, or if this is the default setting, you will need to turn it off. Learn more about AP Isolation here. If the network you want to connect to is hidden (a network that is set not to broadcast its name), you'll need to manually input the network's name and password. Please call your network provider if you don't know the name or password of your Wi-Fi router.
If you wish to connect your Chromecast without using Wi-Fi and have an available Ethernet connection nearby, you might benefit from using the Ethernet adapter.