You should add a startup item to your startup application list, as mentioned by Mladen B.

The command is:

/opt/telegram/Telegram 

Or, if you've installed Telegram via the software center or snap:

telegram-desktop

To start hidden:

/opt/telegram/Telegram -startintray 

There is a -now closed- bug in Telegram using Unity: a sequence of "Open Telegram --> Minimize to tray --> Open Telegram" will show the window.

You can add it to the list of startup applications, by going to Dash and start typing "startup". The "Startup Applications" icon should show up.

Click on it and when it starts, click on the "Add" button to add your app to the list of applications which will start automatically each time you log in. Also you can start it minimized by adding -startintray commandline option:

Sadly none of them are work from the prevoius solutions.

What I did:

1. Created a bash script file with (You have to add the proper path of your Telegram executable. Relative path didn't work!):

   #!/bin/bash
   /home/spyff/.telegram/Telegram &
   
   while true
   do
       xdotool windowminimize `xdotool search --onlyvisible --classname Telegram`
       if [ $? -eq 0 ]
       then
           break
       fi
       sleep 0.1
   done
   

2. Saved somewhere as TelegramStarter.sh

3. Added that script to the Ubuntu's Startup Applications

This start Telegram and if its actually started, minimize it immediately. Tested on Ubuntu 16.04.3 with Telegram 1.1.23.

You could use crontab, and enter this in the terminal:

  crontab -e telegram
  crontab restart

and it might work fine this way.