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