Here is the full list of all 13 BATTLESHIP: Official Edition achievements.
Destroy 500 ships
Destroy 1000 ships
Collect 128 medals
Win 10 games in hard difficulty
Destroy 100 ships
Collect 38 medals
Collect 12 medals
Win a game in hard difficulty
Achieve 50% accuracy in a single game
Get a score of 300 points or more
Rank up for the first time
Hit a ship four times in a row
Win 100 games