There are 4 main Naval missions that need to be completed in order to get this achievement. NOTE: When you visit any of the harbormaster's, these missions will look like crosses in a circle, the other icons that looks like wheels are for the privateer missions/achievement.
The 4 main missions for this are: The Chase, The Rescue, French Involvement, and Biddle's Hideout.
You can upgrade the Aquila prior to doing these missions to have the best chances, but it is not required. I did all 4 with only upgraded cannons on mine.
Investigate disturbances off the coast of Martha's Vineyard.
Limit the ship's damage - 50%
Limit damage from the fort's mortars - 50%
Use mines to eliminate gunboats.
Complete all the constraints in one playthrough
Rendezvous with Amanda near Nantucket and find out what she knows about Nicholas Biddle.
Limit Missile shots 0/2
Sink two ships using their weak points
Limit Wave damage 25%
Limit ally damage 50%
Destroy more ships than your ally
Take out the captain within the time limit 1:00
Take out the Man of Wars in the time limit 3:00
Take out the mast of the main ship in one chain shot
Only Kill Biddle
Side Note: You do not need to 100% these missions in order to unlock the achievement, you just have to complete them. At the end of the last mission, Biddle's Hideout, you will get the achievement. I did this without much trouble or upgrades to the ship.
Some of it just takes a little patience and good timing, as far as taking out the Man of Wars. I used the chain shot to take out their masts so that they were immobile. Once they can't move, they're pretty easy to take out.
The last two missions involve you boarding the ships and taking out either the Caption or Biddle himself, this is also relatively easy. I just ran passed everyone else and found my target and took them out.
Again, not too hard to do, though it may take a few times through the last two missions to get the timing down on taking out the bigger ships.