Derive an expression for time period of a satellite in terms of radius of the orbit?

This is simple if you assume circular motion. The acceleration of gravity from the central body must be exactly equal to the centripetal acceleration required for the circular orbit:

GM/r^2 = v^2 / r
v = sqrt(GM/r) ... orbital speed v in terms of G, M=mass of central body, r=radius of orbit
The period is the time it takes to travel the distance 2*pi*r around the circle at speed v.