From the home screen, press the fight button in the bottom right corner of the screen, on the next screen you will be given a list of players to fight. Pick anyone (preferably very easy or easy) and commence with the battle. Win or lose, you still add one to your total, so try and complete battles on both land and air, and also keep an eye out for bounties too, so you can progress with other achievements too. Hit 1000 battles and the cheevo should pop! Happy fighting!
*EDIT* As Fooga pointed out in the comments, you can just load up a fight, and quit out of the app. This still counts towards the total, and I have confirmed myself that it works. Big thanks to Fooga for a quicker solution! :D