That can be a tricky problem to track down, as it sounds like something your website is doing is causing high CPU usage.
What website software is your server running?
If it’s something like WordPress, Joomla, or Drupal, it could be related to a plugin that the site is using.
You might also want to log into MySQL, and run “show full processlist” to see if there’s any queries that are “large” that this user is repeatedly running.