1. First, you'll need to purchase the METAL GEAR SOLID Pack, if you don't already have it. You can purchase this in the LittleBigPlanet store - the bottom-most option while your Sack Person is using your pod computer. (Make sure to buy the correct pack, because there's also a costume pack available. While dressing your Sack Person up like Snake is definitely cool, it won't help you get this trophy.)
2. You can't just buy the pack and start throwing paintballs around your levels, unfortunately. To the right of the LittleBigPlanet Store is My Content; scroll to this, select it, and find the Metal Gear Solid Pack.
3. You need to complete the following two missions before you'll unlock the Paintinator for your own use: Metal Gear Solid Act 1: Introduction, and Metal Gear Solid Act 2: VR Training. After completing Act 2 (which is really just a Paintinator tutorial level), you'll unlock the Paintinator for use in your own levels.
4. Go to your moon and scroll over an open crater in which to create a level. Select Create. Level theme doesn't matter.
5. Press Square to open your Popit, and select the icon in the lower-lefthand-most corner: the Tools Bag.
6. Scroll two pages to the right, to Gameplay Kits. Scroll down to Character Enhancements.
7. Scroll over the Paintinator Powerup. Press X to select it, and then press X again to place it.
8. Save your level and return to your pod. While looking at your moon, scroll over the level you just made with the Paintinator. Press Square in order to see more options, and then scroll one page to the right.
9. Click Publish. Your trophy will pop as the game publishes your level!