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Basically it's a question of stopping your database server; upgrading the software packages (probably RPMs in this case), and restarting the server. MySQL doesn't (usually) require that a database be exported and reimported when upgrading the server version, so you can (probably) leave the data files in situ.
The MySQL documentation at dev.mysql.com is worth checking for upgrade notes.
How you do this in your particular environment I'm not sure, I'm not familiar with Plesk and Co.