Alerts are regular events in Cloudera Manager just with a special flag. Events are stored in the Event Server internal index, there is no log file to access all previous alerts in text format.
The simplest way to archive what you want is to configure email alert delivery and archive the emails on your mail server. Alternatively send alerts to SNMP management system and store/archive it there.
A third option would be to use a Custom Alert Script where you can process (and then archive) the alerts in the way that is most convenient for you.