They are almost entirely unrelated. When trying to leave the gravity well of a planet, the atmosphere is only a dragging force acting to reduce your thrust. It might be proportional to the surface area of the vehicle, but likely not - I think it's only proportional to the surface area of the "nose" of the rocket. But what's certain is that it's strictly a force that hinders you - in a rocket, all of your thrust comes from the engines, you don't get any boost from the air.
However, even if you're taking off of a planet with no atmosphere, you still have a huge force to deal with - you need to maintain an acceleration to exit the gravity well of the planet, and you need to burn fuel for that. But you also have to carry the fuel you'll burn with you, so the more fuel you have, the more fuel you'll need - this is what the rocket equation codifies.
However, even if you're taking off of a planet with no atmosphere, you still have a huge force to deal with - you need to maintain an acceleration to exit the gravity well of the planet, and you need to burn fuel for that. But you also have to carry the fuel you'll burn with you, so the more fuel you have, the more fuel you'll need - this is what the rocket equation codifies.