Bangai-O HD Missile Fury

Very easy on Level 1. Try not to destroy the blue boxes on the outer edges because those are the "spawners" (where the enemies come from). Stay in the middle of the screen and shoot missiles or counter missiles to keep killing enemies and adding to your combo at the top of the screen. You will see a small blue bar below the combo counter and this is how long you have to kill another enemy / missile in rapid succession. As long as the blue bar doesn't get depleted, your combo will not run out. Once it says 999 on your combo counter, you will get the achievement.

By just hanging around in the middle of level 1 and using "lock in position" and repeatedly bombing, you'll get this fairly easily.

Consider watching some of the high-up replays for level 1, as they all get a 999 combo.