Here is the full list of all 24 Battle Arena achievements.
Die 10 times
Play 10 matches
Play 50 matches
Play 100 matches
Play 200 matches
Play 500 matches
Play 1000 matches
Win 10 wins
Win 50 wins
Win 100 wins
Win 200 wins
Win 500 wins
Die 50 times
Die 100 times
Die 200 times
Die 500 times
Kill 10 times
Kill 50 times
Kill 100 times
Kill 200 times
Kill 500 times
Kill 1000 times
Die 1000 times
Win 1000 wins