Put the mysql jar in the <context>/WEB-INF/lib directory in Tomcat. Application servers typically ignore the classpath since this allows them to create separate running environments for each of the applications thay are running. Any jars placed in the WEB-INF/lib directory automatically get loaded and made available to your application context.
the answer is contained in your question. You need to add the JDBC driver to the lib directory of your web application (I guess it's a jar file?).
Web applications have access ONLY to what's in their own library (WEB-INF/classes and WEB-INF/lib) and to default servlet functionality (servlet.jar or j2ee.jar) and nothing else. They're fully self-contained.