First I'd say you probably want to turn off persistent connections as they almost always do more harm than good.

Secondly I'd say you want to double check your MySQL users, just to make sure it's not possible for anyone to be connecting from a remote server. This is also a major security thing to check.

Thirdly I'd say you want to turn on the MySQL Slow Query Log to keep an eye on any queries that are taking a long time, and use that to make sure you don't have any queries locking up key tables for too long.

Some other things you can check would be to run the following query while the CPU load is high:

SHOW PROCESSLIST;

This will show you any queries that are currently running or in the queue to run, what the query is and what it's doing (this command will truncate the query if it's too long, you can use SHOW FULL PROCESSLIST to see the full query text).

You'll also want to keep an eye on things like your buffer sizes, table cache, query cache and innodb_buffer_pool_size (if you're using innodb tables) as all of these memory allocations can have an affect on query performance which can cause MySQL to eat up CPU.

You'll also probably want to give the following a read over as they contain some good information.

It's also a very good idea to use a profiler. Something you can turn on when you want that will show you what queries your application is running, if there's duplicate queries, how long they're taking, etc, etc. An example of something like this is one I've been working on called PHP Profiler but there are many out there. If you're using a piece of software like Drupal, Joomla or Wordpress you'll want to ask around within the community as there's probably modules available for them that allow you to get this information without needing to manually integrate anything.