Here's the updated version on how-to install Apache Solr 5.x and get it running with Drupal (WITHOUT Apache Tomcat; as Solr 5.x doesn't support Tomcat anymore).
Install Java 1.7
yum install java-1.7.0-openjdk # or aptitude install java7-jdk
Create a symlink to wherever your /etc/bashrc or ~/.bashrc has
export JAVA_HOME=/usr/local/jdk set to.