The reason that the universe didn't become a giant blackhole has nothing to do with the charge of the graviton or self-repulsion. Even in Newtonian gravity, the universe would be fairly wide instead of a giant bulk of matter. This is because things attract each other but they can orbit if they have a transverse velocity component to the direction of attraction.
On the other hand, if you consider the term, charge, as a generalized concept of simplectic geometry and analitic mechanics, then gravity involves a charge, namely, the energy and momentum.
Let me back up a little.
Electric charge is the conserved quantity of the electromagnetic interactions which originate from the local phase symmetry. This is called $U(1)$ gauge symmetry since it has only 1 phase parameter. In a similar way, we have other gauge symmetries in the universe. For instance, there is color charge which is the conserved quantity of the strong nuclear force because of another phase symmetry, $SU(3)$ gauge symmetry. There is also $SU(2)$ symmetry for the weak interaction but it is broken at lower energies than ~200 times the proton mass. These are quantum mechanical gauge symmetries.
Here, a gauge freedom exist because of these symmetries, meaning that an extra field would show up because of the symmetry. For electromagnetism this field is called photon, for strong nuclear interaction it is called gluon, and for weak interaction they are W and Z bosons.
Gravity is also a gauge symmetry but a classical one, just like the Maxwell theory. The symmetry of the gravity is called local translation symmetry, or diffeomorphism symmetry, which means the local shifts on space and time. The conserved quantities (charges) of this symmetry are energy and momentum because energy and momentum would remain the same after shifting in time and position, respectively. The gauge field of this symmetry is called gravitational potential, $\phi(x)$, but since it is classical it is not a particle.
However, it is believed that one day physicist will find a quantum theory of gravity, so the hypothetical gauge particle of this symmetry would be a spin-2 particle, called graviton. It would have no electrical charge, no color charge, or no weak charge. But it would have energy and momentum. So, it may carry a gravitational charge but if it is massless it would follow the null geodesics just like light, and it wouldn't interact with itself.