I'm not sure why it isn't found, are you running a web application (eg Tomcat)?
One way which may fix it is to move the JAR up one directory to C:\Program Files\Java\jdk1.5.0_05\jre\lib\ext rather than in the mysql-connector-java-3.1.12 directory.
This will add the classes as an extension to the JRE and it is an incredibly brute force way to go about things. It's about as public as you can possibly make a library, which isn't necessarily a good thing. (Oh, you can also remove it from you class path since the classes will be discovered and made available in the same way that the Core Java rt.jar classes are discovered.)
But at this stage lets just aim to get it working.