While I appreciated Snapshot103's suggestion, I respectfully disagree on one thing:
You will find the hard badge MUCH, much easier if you begin by putting 2 into height and 1 into fuel. This gives you greater lift at the beginning and a chance to see more green, yellow and purple platforms on the way up.
Then keep moving horizontally in one direction. This gives you a higher likely hood of hitting platforms and makes this much less about luck.
All his other purchase suggestions are spot on, though.
Just to be super clear about Choripan's comment. Don't go through the tube that sucks you UP in the throne room. I misunderstood what he meant by the hamster tube. Seriously. Don't do it! Plus+ This!
The ramp up to the boss is just annoying. It adds nothing, except a grating delay until the showdown. Love the game, but when you do "part 2" I'd love to see this changed.