Firstly, this challenge is not initially available. You can view it from the list of challenges, and it will indicate how many more challenge level completions are required for it to unlock.
For this achievement, you do not need to complete the 'Dr. Wily Forever' challenge with a gold medal, just as long as you can complete it within he initial time limit.
If you need help, or if you are going for a gold medal, you can simply view the leaderboards and select "replay" to see the fastest players complete this challenge. Replicate the strategies as best you can, and you should be able get it after a few tries.
For quick reference, here are the Wily weaknesses:
MM1: Form 1- Fire weapon
Form 2- Electric weapon (with pause buffering for faster completion)
MM2: Bubble Gun (Utilize Autofire for faster completion)
MM3: Form 1: Shadow Star
Form 2: Snake (into the hatch on top of the robot)
MM4: Pharaoh Ball (Charged up)
MM5: Form 1: Arrow
Form 2: Beat
MM6: Form 1-3: Silver Tomahawk
If you are going for gold, my time (to use as a reference) was 7:00:10.
Please comment if you have any questions or additions.