For this achievement you simply need to destroy 1000 enemies with bombs. This is a cumulative total throughout all gameplays and does not have to happen in one playthrough. As the description states you can do this on any difficulty.
You will most likely earn the achievement over time with multiple playthroughs while going for some of the other achievements, but if you want the achievement over and done with quickly, here's a quickish way to do it.
From the main menu go to options and increase the maximum bomb count to 7 and decrease your shield to 1%. Then start a game via stage select and start from stage 8 on practice difficulty as later stages start with more enemies on the screen at once.
When the level starts, spam your bombs like crazy until you have used up all 7 bombs. Now with practice difficulty the enemies do not fire at all, but when you are out of bombs, simply run into an enemy once and you will die with one hit because of your 1% shield from the options menu, simply continue and repeat the process. You can of course just choose to start from the beginning and do it this way, it is entirely up to you, but I personally found this method to be quicker and easier.
The achievement tracker does not update until the end of the scene and the achievement itself unlocks at the end of the scene or level.