This achievement is part of the Title Update for this game.
The easiest method I found to unlock this achievement was to play the Endless game mode and keep playing it. After I reached Level 41-ish this became ridiculously easy and I had no problems to reach combos as high as 3000.
The only tip I can give is to move around A LOT and make sure to pick up most of the pickups that are dropped from certain defeated enemies.
When facing the "bosses" between the levels, I found out that a circular movement worked best. This ensured that I didn't get hit at the same time as I picked up whatever pickups that were dropped. When I had a bunch of enemies in my HUD, I just moved all over the place, again bearing in mind not to get hit while picking up the pickups. Don't focus on the combo at all, it will come naturally. If you focus on the combo, you'll lose focus on everything else happening around you. Simply focus on defeating the enemies and, I'll say it again, picking up the pickups.
Oh, and I'm terrible at Shoot 'Em Up-games. If I can, you can.