I’ve just installed Avast! Free Anti virus and would like to know under the Current Registration will expiry with a given date. Does the product continue to run and receive security updates after the expiry date of registration?
You just need to re-register your copy at the end of the current subscription (with the same email if you want).
Do I need to remove and reinstall the program?
Well I think no… 
No, you do not need to uninstall and reinstall the program after the expiration date.
You will start to get a reminder to re-register 14 days before the expiry date.
So you just follow the same process you did to register initially, from the avastUI, Maintenance, Registration, that interface doesn’t change until you start to get the reminder so you can re-register earlier than that.