As the title says, you are required to surpass your leaderboard score 3 times so the best thing you can do is to collide with a mountain, get 5 points and then waste all of your fuel by press the red circle button on the left of the screen until you run on empty. This will give you a leaderboard score of 5... as you would know.
Do this a second time but hit a mountain twice then hit the red button to get rid of fuel quickly (to record a leaderboard score of 10), and then do it a third time with 3 mountain collisions (to log a leaderboard score of 15).
It's worth mentioning that this achievement is definitely buggy. Running this on my HTC Mozart, I struggled to log a 3rd leaderboard score (I would get game crashes or the leaderboard wouldn't load or log my score).
After some experimenting I can confirm that to log a leaderboard score successfully without the game crashing or not loading the leaderboard/recording your score, you MUST turn off notifications and your location. To switch these two options off, to Lunar Lander's 'Help & Options' button. These two options are the last two options that you can toggle on/off.
Hope that helps