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 1.8
yum install java-1.8.0-openjdk.x86_64 # or aptitude install java8-jdk
Create a symlink to wherever your /etc/bashrc or ~/.bashrc has
export JAVA_HOME=/usr/local/jdk set to.