I guess the only remaining challenge (since there's no difficulty levels) would be to do a 1CC with only the basic, unupgraded Batarang. But that would be pure masochism so I doubt I'll do it.
The key to beating the game is to upgrade the green weapon to max and never lose it (should be at full power by the end of the second segment of stage 1.) Also, a lot of people don't seem to know about the hidden 1-ups; you can get one almost immediately at the beginning of stage 1 by clearing 5 purple guys on the upper deck with a single charged Batarang. And on the following segment, there's a column of those flying helicopter things that will drop a 1-up if you destroy all of them before they move downward. Also, lots of people underestimate how useful kicks are. They've got three major uses: first, the quickest way to move is by holding Down and mashing the jump button. Second, this technique is very good for killing stronger enemies; those big clowns in stage 1, for instance, are pretty much harmless if you just kick them to death. Third, jump kicks are extremely handy for avoiding attacks because they extend your time in the air (a double-jump without the additional height, more-or-less.)
Once you get a maxed-out green weapon and use your kicks a lot, it's just a matter of memorizing enemy patterns. Though, truth be told, I never got good at the Mad Hatter fight; my key to beating him was to take him out with the green weapon before the more difficult phases could begin. That fight can easily use up all of your lives if you have a weapon at suboptimal power; undoubtedly the hardest boss in the game.