To unlock the Challenge Mode, you have to have the 'Wily Program'; you'll need to beat Dr. Wily's Chapter 3 battle in order to gain the item and thus gain access to the Challenge Missions and Endless Attack.

@MegaBossMan : So you mean it will unlock for Dr.Light? (And maybe the other Dr.’s?)

If you unlock the Wily Program, any doctor with a postgame will have access to the challenge missions; I can't exactly remember if it'll show up for doctors who haven't progressed far enough and completed their campaign, but you'll know well enough when you unlock the Wily Program!