Is the expansion of the universe greater than the speed of light? If the rate of expansion of the universe is increasing , won't it eventually become greater than c? Also, after inflation, did the rate of expansion slow down and then start to accelerate again?
After inflation, the expansion of the universe did indeed slow down. During the inflationary epoch (lasting roughly $1 \times 10^{-33}$ seconds), the universe expanded by a factor of $10^{26}$. That's incredible! However, the inflationary epoch didn't last long, and that incredible expansion ended pretty soon after it started.

The universe is, as you said, still expanding. In fact (like you also said), that expansion is increasing. However, we have to be careful when we talk about the rate of this expansion. Currently, the rate of expansion between two objects depends on the distance between them, which is encoded in Hubble's law:
$$v=H_0D$$
where $v$ is the recessional velocity, $H_0$ is Hubble's constant, and $D$ is the proper distance between the objects. This relation proves that objects that are farther away are moving away at a greater speed. We can extrapolate from that that, at a sufficiently far distance, two objects would be moving away from each other at the speed of light or greater! However, this is only true for objects very far from each other. But yes, eventually, any two objects sufficiently far from each other will move apart at the speed of light - and then greater.

Source (for length of inflationary epoch): http://www.universe-galaxies-stars.com/Cosmic_inflation.html

Source (for Hubble's law): http://map.gsfc.nasa.gov/universe/uni_expansion.html