Play as a Kilrathi ship and it'll be just you vs. the meteors. If you play as the other faction (Terrans?) there will be a Kilrathi opponent in the game as well, making this achievement much harder.
Getting to level 20 isn't that hard. It only gets really crazy on the last few levels. Make sure you grab as many powerups as you can to increase your firepower and refill your weapons and concentrate your fire on the meteors which are flashing color (those are about to hit a station). You can lose a few of the stations as long as you don't lose them all. They respawn and and are reset to full health on each new wave so just make sure that at least one makes it. I picked one of the medium ships for this achievement - the heavies are too slow to move around the map quickly and the light ones have lousy weapons. I relied on my guns for most of the meteor killing.
One last thing - if you want to unlock all the ships, you'll need to get to wave 5 on Meteor Storm as each faction. You should see a message at the bottom of the screen when they unlock.