I wonder, how a graphite can be a good conductor of electricity but at the same time, be a bad conductor of heat.?!
You have to be more precise. Can you give values? Graphite is a very anisotropic material. Its thermal conductivity is insanely high in the x-y plane (about 4 times that of copper). But in the z direction, the thermal conductivity is very low, about 2 orders of magnitude less than copper. Something similar happens with its electrical conductivity.
therefore thermal conductivity is directly proportional to electrical conductivity.
Not quite. The so-called Wiedemann-Franz law (which works reasonable well for metals) stipulates that the ratio between the thermal conductivity and electrical conductivity is proportional to temperature (and not merely a constant!). It also assumes that only electrons contribute to heat transfer, not phonons. But keep in mind that this law does not apply "as is" for semiconductors or semimetals.
Also I guess it is not necessary that a good conductor of heat be also a good conductor of electricity because vibrational energy can also be the cause of it like in the case of diamond.
You guessed right. This would hold reasonable well for metals, but not for heavily doped semiconductors (good thermoelectric materials), or insulators. The more the phonons have a non negligible contribution to the heat transfer, the less the statement linking $\kappa$ to $\sigma$ should hold.