Heavy Boots requires that you make it through a level from start to finish without ever letting up on the boost. It can be a little finicky at the beginning of the level so it is best to start holding the boost button down starting on the loading screen. Hold it down the entire level, make it to the end in one piece, and you've got your trophy.
The first mission is your best bet. Hold down the X-button to shoot, the booster, and then just cruise through the mission. The only particularly tricky part are the little circle walls at the end, but they shouldn't be much trouble after a couple of tries.
If you do this, you should also be able to easily get the "High Flier" trophy as well, provided you get through the level without dying, while breaking all of the glass and getting all of the rescue pods. On this level, they are all directly in your path, so it's pretty hard to miss.
Alternatively, this trophy can be done relatively easily on level 3 (Second Nature), which requires you to boost through the entire level to get the "Basic Training" trophy. It is possible to get all three trophies on level 3 by doing this. It is probably easier to get two of the trophies on one level and pick up the last one on the other level.