En principio necesito instalar java:

$ sudo aptitude install sun-java6-jdk
$ java -version
java version "1.6.0_07"
Java(TM) SE Runtime Environment (build 1.6.0_07-b06)
Java HotSpot(TM) Client VM (build 10.0-b23, mixed mode, sharing)

El java home puede encontrarse por ejemplo, con:

$ locate bin/java

En nuestro caso se va a encontrar en:

/usr/lib/jvm/java-6-sun-1.6.0.07/

Creamos el usuario y grupo tomcat:

$ sudo -i
$ groupadd tomcat
$ useradd -g tomcat tomcat

Luego me descargo el binario de Tomcat 5.x:

http://ftp.udc.es/apache-dist/tomcat/tomcat-5/v5.5.27/bin/apache-tomcat-5.5.27.tar.gz

$ cd /usr/local/
$ tar -zxvf apache-tomcat-5.5.27.tar.gz
$ ln -s apache-tomcat-5.5.27 tomcat
$ chown -R tomcat:tomcat apache-tomcat-5.5.27

Las variables de entorno necesarias

$ export JAVA_HOME="/usr/lib/jvm/java-6-sun-1.6.0.07/"
$ export CATALINA_HOME="/usr/local/tomcat/"
$ export PATH="$PATH:/usr/local/tomcat"

Inicio el tomcat:

$ cd /usr/local/tomcat/bin
$ sh catalina.sh run

Using CATALINA_BASE:    /usr/local/tomcat/
Using CATALINA_HOME:   /usr/local/tomcat/
Using CATALINA_TMPDIR: /usr/local/tomcat/temp
Using JRE_HOME:       /usr/lib/jvm/java-6-sun-1.6.0.07/
.
.
.
INFO: Server startup in 1732 ms

Chequeo la conexión:

$ netstat -nap |grep 8080
(No todos los procesos pueden ser identificados, no hay información de propiedad del proceso
 no se mostrarán, necesita ser superusuario para verlos todos.)
tcp        1      0 127.0.0.1:36260         127.0.0.1:8080          CLOSE_WAIT  9365/gvfsd-http 
tcp        1      0 127.0.0.1:36259         127.0.0.1:8080          CLOSE_WAIT  9365/gvfsd-http 
tcp6       0      0 :::8080                 :::*                    ESCUCHAR    21860/java      

Otro escaneo de puertos:

$ nmap localhost
Starting Nmap 4.53 ( [http://insecure.org] ) at 2009-02-06 12:14 CET
Interesting ports on localhost (127.0.0.1):
Not shown: 1705 closed ports
PORT     STATE SERVICE
8080/tcp open  http-proxy

Creamos un script de inicio:

$ vim /etc/init.d/tomcat
#!/bin/sh

#
# Tomcat init script of catalina.sh using tomcat user
#
#  

# Funciones
. /lib/lsb/init-functions

# Variables de entorno
export JAVA_HOME="/usr/lib/jvm/java-6-sun-1.6.0.07"
export CATALINA_HOME="/usr/local/tomcat"
export PATH="$PATH:/usr/local/tomcat/bin"
export TOMCAT_BIN=$CATALINA_HOME/bin
export TOMCAT_USER=tomcat
#pid=`ps auxww | grep catalina | grep -v "grep" | awk '{print $2}'`
#pid=`pgrep -u $TOMCAT_USER -f catalina`
pid=`jps |grep Boot| awk '{print $1}'`

tomcat_start(){
                log_daemon_msg "Starting tomcat"
                if [ $pid > 0 ]; then
                        log_success_msg "Tomcat already running"
                else
                        su -c "$TOMCAT_BIN/catalina.sh start" tomcat
                log_end_msg 0
                fi 
}

tomcat_stop(){
                log_daemon_msg "Stopping tomcat"
                if [ $pid > 0 ]; then
                        su -c "$TOMCAT_BIN/catalina.sh stop" tomcat
                        log_end_msg 0
                else
                        log_success_msg "No Tomcat process found."
                fi
}

case "$1" in
        start)
                tomcat_start
                exit $?;;
        stop)
                tomcat_stop
                exit $?;;
	 force)
                if [ $pid > 0 ]; then
                        kill -9 $pid
                fi
                log_success_msg "Tomcat killed OK."
                exit $?;;
        status)
                if [ $pid > 0 ]; then
                        log_success_msg "Tomcat is running (pid=$pid)."
                else  
                        log_success_msg "Tomcat is not running."
                fi
                exit $?;;    
        restart)
                tomcat_stop
                sleep 10
                tomcat_start
                exit $?;;    
  	*)
                echo "Usage tomcat5 start|stop|restart|status|force"
                exit 1;;
esac

Editamos la configuración del runlevel:

$ sudo chmod o+x /etc/init.d/tomcat
$ sudo update-rc.d tomcat defaults

 Adding system startup for /etc/init.d/tomcat ...
   /etc/rc0.d/K20tomcat -> ../init.d/tomcat
   /etc/rc1.d/K20tomcat -> ../init.d/tomcat
   /etc/rc6.d/K20tomcat -> ../init.d/tomcat
   /etc/rc2.d/S20tomcat -> ../init.d/tomcat
   /etc/rc3.d/S20tomcat -> ../init.d/tomcat
   /etc/rc4.d/S20tomcat -> ../init.d/tomcat
   /etc/rc5.d/S20tomcat -> ../init.d/tomcat

$ sudo /etc/init.d/tomcat start