The positron states and annihilation characteristics are calculated for different phases of carbon: pure and potassium-doped solid C60, diamond and graphite. In solid C60 with the FCC structure the positron is concentrated in the interstitial space between the C60 molecules. For the different carbon phases the calculated positron lifetimes reproduce well the experimental results in a wide range from about 100 ps for diamond to 400 ps for fullerite. The effects of interstitial dopants on the positron distribution and lifetime are studied for the solid C60. Finally, the positron characteristics for the compound KxC 60 are predicted and found to be sensitive to the potassium concentration.