Startup Crontab
This simple method will simply start your TeamSpeak server instance when your server has finished booting. Consider using a Systemd Unit if your system supports it for more advanced features.
Pretty much all Linux distributions support this method as crontab is installed on most distributions by default.
Add a line like the following to your crontab (run crontab -e to get to the crontab editor on your system)
@reboot /home/teamspeak/teamspeak3-server_linux-amd64/ts3server_startscript.sh restart