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