I unlocked this on the same run as Naked Specter, which I would recommend doing.
For this run, you want to waste as little time collecting Red Skulls as possible.
The level order should start with Flying Machine to get the Judgment Rush to speed through some sections and the second level should be the Lich Yard to get the Skeletal Sniper, which allows you to defeat bosses easily.
After that, I would recommend completing the Clockwork Tower to get the Barrier Lantern, which is always useful.
You should get the Rail Mail
as quickly as possible so that you can save time on the levels.
The most important tip here is that if you feel that you've died too many times on a level, hitting and returning to the main menu will reset the timer to whatever it was before you started the level