Take no prisoners!

Kill the crew of 20 ships by sector 6 in the Mantis Cruiser.

This one is probably the easiest achievements to get. You need to kill the entire enemy crew of each ship to be killed and leaves the enemy ship intact. Destroying to ship with crew inside doesn’t count.

"By sector 6" means the crews have to be killed by the end of the sixth sector, not by the beginning of it.

Avast, ye scurvy dogs! 

Kill 5 enemy crew in a fight without taking hull damage or losing a crew member while using the Mantis Cruiser.

The easiest way to do that achievement in when you encounter a ship with Clone Bay system. Disable their weapon or if you have enough shield to prevent them from doing you damage. Board the ship and kill everyone with your mantis. wait for them to re-spawn and kill them until you get the 5 kill.

Another way to get that achievement it’s when people board you at the same time you are attack by a ship event though there no teleport on the ship. Finish the boarder then board the ship. If you’re lucky like me there will be 3 boarder and 2 on the ship.

Besides a fight against an enemy ship, this achievement can be won from a shipless boarding event with five or more attackers.

Note that Boarding and Anti-Personnel drones can be used to help unlock this achievement.

Battle Royale 

While using the Mantis Cruiser, kill the last enemy with your last crew member on their ship.

This is the hardest of the three mantis achievement to get. I manage to get it with the layout B. I’ve sent my 2 mantis while fighting a nearby sun (not recommended) and their crew was harm by the fire, but one of my mantis had fallen in combat, the other one manage to kill the last one while having just a few hp left.

Note that this achievement can be unlocked with the assistance of a Boarding drone, has long it doesn’t get the kill. The final blow must be dealt by the crew member sent to board the enemy ship.

You can cheat to achieve this one. In the first sector board an enemy ship and kill all the crew except one and bring the last crew to almost no health, then teleport back your crew back. Then you put one sole crew member in the teleporter and kill your other crew members with asphyxiation. Finish the last enemy off with your remaining crew member.