First, add the client as a user to your workspace. Learn how to add a new user to your shared workspace. Do make sure that they accept the invitation so that he/she can access your shared workspace.
Once in the shared workspace, your client simply needs to login and head to Ahrefs Alerts (Backlinks Alerts, New keywords Alerts, Mentions Alerts) to set up the alerts by themselves so that they can receive respective alerts to be sent directly to their inbox.
Backlinks, New Keywords and Mentions Alerts are independent alerts. They are tied to the alert creator only, which means that they are private alerts that are not shared among different users within the same workspace.
Alert emails consume credits per month. Learn more about what actions consume credits.