Basically, check:
1) If your subscription allows the update right now (i.e., if it is not expired).
2) Check you firewall settings and allow avast.setup to connect. If you're not sure, just turn off your firewall (to be sure the update failure is not due to it)
3) Check your proxy settings into avast!:
Left click the avast icon > Settings
Update (Basic) > Details > Proxy
Make sure your browser is not set to "Work Offline" (this option is generally in the File menu). If it doesn't help, try switching the proxy settings from "Auto-detect" to "No proxy" (I'm assuming you're not connecting to the Internet via proxy).