There are 3 challenges per level. They are for not dying, for scoring a a certain damage bonus, and for beating the level in par time. To get these challenges you must unlock the level first. While highlighting it, go to down by pressing . Now go to where there is a well and a guy whom is the bookkeeper. He will ask you if you want to place a bet. Talk to him and he will ask you to pay gold to do one of the challenges. The gold required raises by 100 each act. You can only do one at a time, meaning you must complete the levels a minimum of 3 times with the same character, actually beating the challenges asked. This gives you a full 2 minimum extra playthroughs.
You can easily get the time limit and no death one by leveling up your character very high and completing the game, then going back to do them. The damage one is a bit trickier and you may have to equip lesser weapons because the damage bonus is based off of combos, and if you kill an enemy in one shot you cannot combo them. Using weaker weapons on earlier levels, you will need to juggle enemies to gain the score. The score is actually 6,666 on every level. Use the combo then up on + to knock the enemy into the air. Jump and press to juggle the enemy and relaunch it into the air. Even when dead, you can still gain the score off the enemy. This can be repeated over and over.
This solution found in this thread here: Score Challenge Thread by ponypo2001 is extremely accurate in telling you how you need to accomplish the score challenges on the early levels with your high level character.