To be clear, you must have less than 7 cards in your deck when you finish the run.
Prioritize side missions that allow you to remove a card.
If you have AI White Candy equipped, it allows you to remove a card for 200 gold, which is a nice extra.
While collecting your rewards, don't get any cards unless they are essential for your deck build.
As mentioned by djbobas on TrueAchievements (
https://www.trueachievements.com/a317803/lucky-777-achieveme...), a great deck for this achievement is Claire's Vampiric deck.
Vampire and
Blood Lust give you a steady supply of Vampiric. Use
Powerful Strike/
Vampiric Strike to get rid of your enemies and heal yourself at the same time.