Probably one of the easiest stages to do is Wily castle 5. If you want to try to get it on an earlier stage, try going on Pump Man stage.
Thanks to KMetalmind for this tip: Another good stage is the Commando Man one. If you take the upper path, it should be easy to finish the game without getting hit. The only enemies who usually can make you lose are the Mets (enemies that hide in their hat and shoot at you). If you are careful with the bombs and with the bugs, it´s a piece of cake :)