As @ACuriousMind pointed out: it is the particles momentum that goes against infinity its rest mass $m$ is a scalar: a fixed particle property that does not change.
To your question about "gravitational mass": in General Relativity (GR) this is a very very delicate point. The concept of a "gravitational mass" is not trivial in GR and one needs to specify very precisely what one understands as gravitational mass. First of all "mass" is not governing the curvature of space: the energy momentum tensor is.
So if you are interested in the gravitational field of a massive particle moving very fast one would need to solve the field equations with the energy momentum tensor of this particle. You can NOT use lets say the Schwarzschild metric (SSM) and put in the "relativistic mass" of your particle. First the concept of relativistic mass is not a good one 2. the SSM does not hold at all for moving particles/sources. It holds only outside a static, spherical symmetric object.
I did not find a reference for the gravitational field of a relativistic moving particle but one can for sure not use the SSM. The gravitational field will depend on the particles four-momentum, it will be non static and not spherical symmetric, since the direction of motion is an axis of anisotropy. Maybe one can work out some spatial axis-symmetric solution for that problem choosing a coordinate system in which the four momentum becomes $p^\alpha=(p^0,0,0,p^3)$ but I have not found/ calculated one. Apart from that I would assume that even at relativistic speeds the curvature caused by single massive particle will be relatively small, since even at very high speeds the energy/momentum of such a particle would be rather small.
This gravitational mass against infinity picture does not hold: its all about the energy momentum tensor and you will not get this one to infinity with just relativistic speeds.
That being said one more point towards "gravitational mass" in GR, to be specific in the SSM. In the SSM the "gravitational mass" comes up in form of a integration constant and from the Newtonian limit we see that this constant can be identified with the classical/Newtonian gravitational mass. This is one special case where classical gravitational mass has a meaning/equivalent in GR. It is one of very few cases where the concept holds some meaning in GR. In general this is not the case and one can define a multitude of "masses" (integrals over very different objects).