You must do the suicide mission after your crew are kidnapped, or they will perish at the start of the mission. Also, all the teammates should be loyal.
To get the "No One Left Behind" achievement get all upgrades for your ship after asking everyone about an upgrade. One way to achieve "No One Left Behind" is for Tali unlock the door, Jack use the biotic barrier, Miranda run the diversion, have Grunt escort the Doctor, and have Mordin in your squad for the final battle.
Another way is to use Tali unlock the doors, Samara use the biotic barrier, Jacob run the diversion, have Garrus escort the doctor, and have Thane and Jack in your squad for the final battle. It is important that whoever is in the final battle with you should be loyal, since they might not catch you during the end of the escape sequence.
Do not have Zaeed run the diversion, since he dies at the end of that segment.