Calculate the Lorentz factor gamma.

The speed of the particle is given as $v = 0.6c$.
The Lorentz factor $\gamma$ is given by the formula:

$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Substitute the given speed $v = 0.6c$:

$$\gamma = \frac{1}{\sqrt{1 - \frac{(0.6c)^2}{c^2}}} = \frac{1}{\sqrt{1 - \frac{0.36c^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.36}}$$ $$\gamma = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25$$

The relativistic kinetic energy $KE$ is given by the formula:

$$KE = (\gamma - 1)mc^2$$

Substitute the calculated value of $\gamma$:

$$KE = (1.25 - 1)mc^2$$ $$KE = 0.25mc^2$$

Comparing this result with the given options:
A. $0.18mc^2$
B. $0.22mc^2$
C. $0.25mc^2$
D. $mc^2$

The calculated kinetic energy matches option C.

The final answer is $\boxed{\text{C. } 0.25mc^2}$.