On the MKF forums, there is a list of optional challenges that can be completed resulting in a trophy award. These challenges are not needed to get the "true ending" to MKF, but merely for fun. New ones are constantly created by forum members.
List of Challenges
The Airship of Hate: Complete Airship Armada without getting hit, using any character. Completed using Mario (no powerups), by Genisi.
Structural Integrity: Defeat Mouser, but don't let any of the breakable bricks get destroyed, using any character. Completed using Luigi, by Target.
Size Doesn't Matter: Defeat all the bosses as Small Mario, not using any powerups. Ongoing challenge run by Genisi.
Retro Rematch: Defeat King Totomesu without getting hit and without using projectiles as any character. Completed using Mario (no powerups), by Genisi.
Fly Swatter: Clear King Watinga's Revenge without getting hit, using any character. Completed using Mario (no powerups), by Genisi.
We Have Much More Important Business: Clear The Library without hitting Guilty Spark. (Note, this may require bringing a flight powerup into the level beforehand so you can complete the run section) Completed by lookw: part 1: http://www.youtube.com/watch?v=O23C9x1TPhc part 2: http://www.youtube.com/watch?v=WRBJ0wr4H5w
No Thanks, I Brought Mine: Defeat the Cyberdemon without having to resort to using the rocket launchers in the arena. Completed by ZeroFollower0 using Luigi.
Bomb Phobia: Defeat Mouser without using any bombs (this requires a projectile powerup to complete). Completed using Luigi, by Target.
Moonshines: Collect the 3-Up Moon in the chase section of Skyscraper Domain Completed by lookw, using Tails (July 12, 2008)
Unidentified Flying Hedgehog: Clear the RUN!! section of The Library using Sonic. Judgespear completed this challenge when he revealed Sonic, actually. Also completed by lookw: http://www.youtube.com/watch?v=HWz-bl3P7Lw
Bug Exterminator: Clear King Watinga's Revenge without using the Lakitu Cloud with any character. Completed using Mario, by lookw.
Clown Crusher: Defeat Calliope The Clown without getting hit as any character. Completed using Mario (no powerups) by Genisi.
Feel The Sunshine: Clear three stages in a row using the Tails Doll suit (as Tails of course) without getting hit. Completed by lookw.
Super Virus Bros: Clear Corpse of the Behemoth without getting hit by anything using any character. Completed by lookw using Tails.
Good Samaritan: Clear any stage without killing any enemies, using any character. Completed by Genisi using Small Mario in levels 1-1, 1-2, and 1-S1. Completed by lookw using Tails.
Heart Attack: Defeat the Necromancer only by throwing the red cells he spawns from the Heart. Completed by lookw: http://www.youtube.com/watch?v=PxU3g8n0150
Hammer Bros Impersonator: Clear any level using Mario only killing enemies(and bosses) using the hammers from the Hammer Bros. Suit. Completed by lookw.
"The Foot" of Judgement: Beat the entire Library without using any projectiles, POW blocks, grenades, or kickable infection forms. No checkpoints, of course. Completed by lookw: part 1: http://www.youtube.com/watch?v=lmJsOz6Ewuo part 2: http://www.youtube.com/watch?v=oq8CDBxyg9I
What Would Chuck Norris Do?: The above without any use of the Spartan Powerup (must begin the level without any powerups or rings as well). Completed by lookw: part 1: http://www.youtube.com/watch?v=alz2-nPkZBw part 2: http://www.youtube.com/watch?v=hWKk22VQe8M
Flak Insanity: Defeat King Totomesu using only Nokobons kicked straight up into the air Completed by lookw using Tails.
Stampede of Doom: Survive Buffalo Evasion using any character, and without getting hit. Completed by Kilometers Persecond using Sonic.
Stampede & Slide: Using Big Mario or Big Luigi only, clear Buffalo Evasion without getting hit. Hint: You MUST slide to avoid the falcons.
Medieval vs Modern Warfare: Beat Scandinavian Campaign using Arthur's weapons only.
Conscientious Objector: Complete Finland Battlefield without personally killing any enemies. YOU YOURSELF cannot defeat any enemies.
Bridge Knight: Beat the Bridge Zone as Arthur without taking damage.
Ghostbuster: Defeat Big Boo at the end of Boo Mansion before he makes 3 complete circles around the arena.
Charged Wood: As Mario, you must complete Wood Man's Forest while holding a chargeshot the whole way. You may NOT release the chargeshot and are only allowed to kill enemies through the buster shots that appear during a spinjump (you can't actually spin jump on enemies). This stops when Woodman appears; anything goes when you actually fight him.
Rambunctious Hedgehog: Playthrough an entire level with Sonic, while running, without stopping.
Funky Highway: Beat Journey to Funkotron without getting hit and without killing any enemy.With any character and without going to the shop for items.