Mass doesn't increase... not exactly. Your inertia increases, because the effects of relativity require that more energy is required to accelerate an object as it approaches the speed of light.
I'm not a physicist, but here's what I understand about Relativity:
Due to the light-speed limit, as an object approaches the speed of light, all of its internal motion has to be redirected toward the direction the object is moving. AT the speed of light, the object can have no other internal motion: electrons can't orbit their atoms. Atoms and molecules can't move around in fluids. Heat and other energy can't move between components of the object. Even the molecules themselves have to stop vibrating.
Time dilation is what you get when that effect starts to kick in: you are forcing the subatomic particles in the object to stop moving along other paths and move in the direction of the object itself. As more of the object's internal motion is wrapped up in moving along that straight line, less is available for chemical reactions and the exchange of energy - kinetic energy included.
This means that you have to exert greater and greater force, relative to the mass of the object, to accelerate the object closer to the speed of light, since you're not just accelerating the object itself, but you're decelerating all of the object's internal motion.
None of this causes the object to increase its gravitational pull against other objects; it just increases the object's inertia.
So the speed of light limit makes an object's inertial mass increase, but not its gravitational mass.