You get a Champion medal for completing all 10 stages in a world, including the bonus and boss, without needing a continue (dying is fine, as long as you still have lives left).
Be aware that to get this - except for your very first play - you need to specifically "go for the champion medal" by pressing X on the stage select screen. You'll know you're on track if you get a popup asking "erase your clear records for this world?", and then another after each stage saying "trying for the champion medal".
Do that 9 times, and you have the achievement.
If you're struggling with any of the stages, check out this video (not mine):
You can also try practicing levels in stage select, then starting from an earlier, easier world to build up lives.