I would like to offer a slightly different strategy than what is offered above. I still recommend using the level select instead of signing a new pact, but I used a slightly different set of tarot cards.
Tarot cards: For Gold Cards, I used Iron Will for invincibility, Fury for more damage, and Rage for even more damage. For Silver Cards, I used Mercy to allow for more Gold Card uses and the DLC tarot card that gives 666 ammo for all guns. (Sorry, I could not remember the name, if anyone wants to submit that information that will be very helpful. Thanks.)
Weapons: I used the shotgun almost exclusively. With nearly infinite ammo you can freeze many enemies then blast them away no matter how far away they are. The Shotgun can be used almost like a sniper rifle this way. Just shoot the weaker baddies and freeze and shoot the stronger ones. This lets you take out almost anyone in the game with two shots, no matter how strong they are. For bosses, you just want to use the rockets and chaingun with your gold cards to take them out fast.
You should do this achievement after you have done all the other campaign achievements so that you know where secret areas are and you know when to use your gold cards.
Also be sure to hoard souls as much as possible, because the demon form will come in handy at taking out some of the harder encounters.
Using the shotgun all the time, you can just sit back in a corner and let the enemies come to you or keep walking backwards and popping them off as you go.
When you do use your gold cards, I suggest using a rocket launcher, shuriken gun or machine gun to deliver the most damage to the most enemies in the shortest amount of time.
For the last boss, only use your gold cards when he is vulnerable (i.e. when he is not green). the rest of the time, just bunny hop around and collect health and armor.