Considering that you are using the correct version of the MySQL Connector/J jar, you need to add the jar both to your CLASSPATH environment variable and also to TOMCAT_HOME\lib folder.
As per the documentation available here, you need to have these settings done so as to work with MySQL JDBC connectivity.
You can set the CLASSPATH environment variable under Unix, Linux, or OS X either locally for a user within the user's .profile, .login or other login file, or you can also set it globally by editing the global /etc/profile file.
For example, add the Connector/J driver to your CLASSPATH using one of the following forms, depending on your command shell:
# Bourne-compatible shell (sh, ksh, bash, zsh):
shell> export CLASSPATH=/path/mysql-connector-java-ver-bin.jar:$CLASSPATH
# C shell (csh, tcsh):
shell> setenv CLASSPATH /path/mysql-connector-java-ver-bin.jar:$CLASSPATH
For Windows platforms, you set the environment variable through the System Control Panel. To use MySQL Connector/J with an application server such as GlassFish, Tomcat, or JBoss, read your vendor's documentation for more information on how to configure third-party class libraries, as most application servers ignore the CLASSPATH environment variable.
If you are developing servlets or JSPs, and your application server is J2EE-compliant, you can put the driver's .jar file in the WEB-INF/lib subdirectory of your webapp, as this is a standard location for third party class libraries in J2EE web applications.