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