Do I need to cancel my subscription (unsubscribe) or something like that in order to be able to re-installing the antivirus in the new pc? Any help would be appreciated. Thanks in advance.
No, just remove it from the old system and install it on the new one.
You mean removing by simply deleting, no cancelling or unsubscribing?
By removing, e.g. uninstalling of the old system, nothing else.
When you install on the new system link it to your my.avast.com account and that should assign the license/subscription to that installation.