You cannot. At least according to General Relativity, without somehow adding in some quantum gravity effects. We don't know enough (or anything definitive) about quantum gravity, but it still might not make any difference outside the BH. It could affect what happens as you get inside the black hole and close to the singularity (of course we don't know this either), but you're staying outside
First, by Birkoff's theorem if you somehow manage to set up a spherically symmetric (say) thin shell of very massive matter OUTSIDE the black hole (BH), it will have no gravitational effect inside the shell.
Second, of course that shell is likely to instead just collapse into the BH, assuming it is a pressureless ideal gas. If it resists it through pressure but no heat or dissipation it'll still collapse. If it has collisions in the shell and some real matter dynamics it'll form quite a maelstrom of radiation and probably a lot worse, but it will not break up the BH. If anything lots of it will be absorbed, and because there is always some rotation it'll probably form a rotating ring that keeps on radiating and kicking off particles and imparting angular momentum to the BH.
Third, if the shell is massive enough, or if it is a semi solid sphere that heavy, it'll just collapse on its own anyway, if it is inside a not-huge radius, say solar sized or less. It'll merge into the BH, and what does not merge in will eventually be radiated away or kicked off as in the second point above.
Fourth, even if you somehow configured a distribution of matter or energy outside the BH that somehow, say on one side of the BH so it didn't have to be symmetrical, could somehow affect the metric (i.e. spacetime curvature) in a way that it reversed (or say just changed) the BH curvature outside the BH, the only thing that could possibly happen to the horizon would be to grow or start rotating. There would be no way for the horizon area to get smaller, or have the BH break into two piece, because BH thermodynamics says that the entropy, and thus the area, must grow or stay the same.
All of these cases are pretty unrealistic as it is not clear how you could assemble such a massive distribution of mass-energy. Most likely it would have to evolve with the star that eventually has to collapse into a BH. We know that stars bigger than about 3-10 solar masses can not do anything except break up before they form a BH
Hard to manipulate gravity, you need lots of mass-energy, and then it tends to evolve by its own equations.