Priority Assignment are received from your Astropath on the command bridge once you've progressed a certain distance through the Season 1 campaign. Speak to the Astropath to start a Priority Assignment, which consists of a sequence of missions in a row. You'll have to speak with him again after each mission to start the next mission in the assignment, or bring up the RT menu on the command bridge and select "Continue Assignment".
I've had Priority Assignments that range from a single mission, up to six missions, and it seems to have little to do with the choices you make along the way. You'll be asked to make certain decisions as you progress, and given the opportunity to buy certain single mission perks as you go along.
For the achievement, you must complete every mission in the assignment without dying during any of them. The difficulty of the Priority Assignment is generated based on your Power Level at the time you take the assignment. So the easiest way to do this is to unequip most of your gear until you get down to a Power Level of around 100. Go speak to the Astropath and a Priority Assignment will be generated with a difficulty of 100 or so. Before pressing the button to actually begin the mission, go into the Pause menu and your inventory, and re-equip all your gear. Now you can blast through all the missions as a way overpowered killing machine. The difficulty level of each successive mission will increase slightly, but it will get nowhere close to your power level.