How can you create non-centripital artificial gravity and apply it to propel a spacecraft by riding the gravitational space time warp?
Nobody knows! Various things have been written and various people have thought carefully about the problem, but ultimately all the ideas fall down on the need for things we don't have, wouldn't know how to manipulate or control, and may not actually exist.
If you could warp space on a whim, you might be able use it to drive spacecraft around, create stealth spacecraft, create defense shields, artificial gravity for the inside of the ship, and maybe even stasis fields.
The sorts of handwavium you'd need include cosmic strings or perhaps some form of tame phantom energy.
This drive uses the gravity's warped spacetime plane to move the craft. By moving the center of the artificial gravity in the direction in which you want the spacecraft to travel, you can steer the spacecraft, and by increasing or decreasing the gravity level, you can accelerate and decelerate relative to the gravity strength.
NASA's Breakthrough Propulsion Physics Program called this a Pitch Drive. It was one of several warp drive ideas they took a look at. Its requirement for magical handwavium and physics that may not be possible seems insurmountable.
Theoretically, by electromagnetically charging mercury, it is possible to create a non-centripetal gravity field, and if the mercury can spin fast enough, this artificial gravity should be able to overcome any gravity.
This is not a theory. This is pseudoscience and technobabble... it'll fit in OK in relatively soft scifi, but that's about it, I'm afraid. I'm of the opinion that this sort of detail should be avoided in any story, because it presents a way for your science to be clearly and unambiguously wrong.
Conserve detail for where it is needed.
If we could learn how to create an artificial gravity field that is stronger than that of a black hole for example, the craft would travel faster than the speed of light
I suspect that gravity field so strong that things fall into it at superluminal speeds is a concept that simply doesn't make any sense. Physics breaks down at a singularity. It probably isn't a place you'd want to visit.
Nobody is quite sure if a warp drive could actually allow superluminal travel... various bits of analysis have been done on the Alcubierre metric and related things, and it has been variously declared to be impossible or impractical (and something that's even more impractical than a warp drive is a pretty seriously unlikely thing) and even if it were possible you open up the doors to time travel and causality violation and paradoxes and all manner of bad things.
Obviously, in your own setting you can handwave your own solutions like chronology protection conjectures and privileged reference frames and so on, and remain relatively hard scifi.
Remember that even when time travel is forbidden, reactionless drives can still represent an existential threat to any spacefaring civilisation as they make the manufacture of planet-smashing relativistic kill vehicles realistic and practical. Consider Burnsides' Advice:
Friends don't let friends use reactionless drives in their universes.
Hard scifi interacts very badly with FTL and reactionless drives. With soft scifi you can wave your hands as much as you like to make your universe interesting but not immediately fatal to everyone in it. Worth considering.