The mass of a particle does not increase as its speed increases. This concept comes from trying to fit the relativistic equations for momentum to the Newtonian equations, which you can't do.

According to Einstein, the momentum of an object moving at speed v is:

p = γ m v

where γ = 1/sqrt(1-v2/c2). Trying to fit this to Newton's expression for it gives the result that rabidcow quotes. Although this works for 1-D motion, if you accelerate the particle in any other direction but parallel to v, you need to change the mass in a different way (for a example a factor of γ3 is needed if you accelerate perpendicular to the motion). The variable mass concept is a helpful idea at first, but only works in very limited conditions, and will get you into trouble if you try to apply it further.

Hence the reason an object cannot be accelerated beyond c should be explained in terms of momentum, not mass. Newton's Second Law of Motion, as he originally phrased it in terms of momentum, still holds, providing the above definition of momentum is used, and requires an infinite force to accelerate beyond c.

Alternatively, the explanation can be done in terms of the infinite amount of energy that must be given particles to get them to reach c (assuming they have mass). You have to use the general form of Einstein's equation:

E = γ m c2

which works for any speed, not just the rest mass energy. Again, we could have done the m = γ m0 trick, but it's unclear which 'mass' is being referred to, and is best avoided for the reason I put above.

Thanks to tdent for corrections