TLDR: Keep the crew alive and keep playing missions to earn XP until you reach level 6. Immediately select a second skill for at least one of your crew. Continue until both skills reach level 12.
Crew members gain experience during missions, through enemy kills and completing objectives. During the mission the XP gained can be seen in the top right. Obviously, if the crew member dies or the ship is destroyed, you'll start with a new crew at a lower level, scaled to your progress.
Each crew members current level can be seen in mission, above their name next to their skill icon. The maximum is level 12. While only a single crew member needs to reach that level in both skills for this achievement, you'll want all your crew to be levelling up for the abilities that will bring.
At level 6, the Comms Officer unlocks a skill called Performance Analysis. This doubles the XP earned for a time and is incredibly useful. Get in the habit of activating it whenever you're heading into a fight.
Once they reach level 6, a crew member can learn a second skill. You need to get both skills to level 12. Along the way, you'll also unlock:The easiest way to go about this it to play on the very easiest settings. At the Mess Hall on Athena Base, press and select the following:
Tagging Mode - Automatic
Difficulty - Casual
Crew Size - Increased (gives and extra Weapons Officer)
Slow Time - Unlimited
With these settings, it's actually fairly difficult to lose a crew member or the ship. Early on, I put the 3 Weapons Officers and Security Officer on the guns. The Engineer stayed at their station.
Once I hit level 6, I trained as follows:
Captain - doesn't matter (stays on station),
Comms Officer - doesn't matter (stays on station),
Engineer - Security Officer
Security Officer - Weapons Officer,
1 x Weapons Officers - Engineer
2 x Weapons Officer - 2 x Security Officer
(To learn a secondary skill, tab across to Training at Athena Base. Select each character and hit
to Secondary Skill. Not all characters can choose every skill.)
This gave me a Weapons Officer for each gun, an Engineer who did all the running about and repelling boarders, a backup Engineer and some extra muscle to fight boarders if required.
As mentioned in the comments, get this done before starting the final mission, "Save The Earth".
I hope this helps!