You will need root privileges for any the following. To get root, open a terminal and run the command

sudo -i

and the command prompt will change to '#' indicating that the terminal session has root privileges.

Alternative #1: Add commands to /etc/rc.local

vi /etc/rc.local

with content like the following:

# This script is executed at the end of each multiuser runlevel

/path/to/my/script.sh || exit 1   # Added by me

exit 0

Alternative #2: Add an Upstart job (for systems older than 15.04) (not recommended)

Create /etc/init/myjob.conf

vi /etc/init/myjob.conf

with content like the following

description     "my job"

start on startup

task

exec /path/to/my/script.sh

Official statement from upstart website -> "Project is in maintaince mode only. No new features are being developed and the general advice would be to move over to another minimal init system or systemd."

Alternative #3: Add an init script (obsolete)

Create a new script in /etc/init.d/myscript.

vi /etc/init.d/myscript

(Obviously it doesn't have to be called "myscript".) In this script, do whatever you want to do. Perhaps just run the script you mentioned.

#!/bin/sh

/path/to/my/script.sh

Make it executable.

chmod ugo+x /etc/init.d/myscript

Configure the init system to run this script at startup.

update-rc.d myscript defaults