Recently for the first time I was able to easily get Tomcat up and running on SSL without feeling the need to rip my hair out. I was able to copy a config from a co-worker. I'm not sure why hours of Googling wasn't able to give me an easy answer. Here's the easy answer!
Comment out the standard http connectors (usually 8080)
<!--
<Connector port="8080" protocol="HTTP/1.1"
connectionTimeout="20000"
redirectPort="8443" />
-->
Your SSL connector should look like this:
<Connector
port="9443" minSpareThreads="5" maxSpareThreads="75"
enableLookups="true" disableUploadTimeout="true"
acceptCount="100" maxThreads="200"
scheme="https" secure="true" SSLEnabled="true"
SSLCertificateFile="/etc/ssl/certs/ssl-cert-snakeoil.pem"
SSLCertificateKeyFile="/etc/ssl/private/ssl-cert-snakeoil.key"
clientAuth="false" sslProtocol="TLS" />
Of course you'll want to customize this but the beauty is, if you are used to dealing with standard Apache type certs, this is plug and play.
Edit: This is my second instance on one host which is why I chose 9443 instead of 8443. If you have other connectors you may have to modify their redirectPort setting.