Thermal conductivity $k$ is defined by Fourier's law of heat conduction:

$Q = kA\dfrac{\Delta T}{\Delta x} \cdot t$

$\Rightarrow k = \dfrac{Q \cdot \Delta x}{A \cdot \Delta T \cdot t} = \dfrac{\text{J} \cdot \text{m}}{\text{m}^2 \cdot \text{K} \cdot \text{s}}$

$= \dfrac{\text{J}}{\text{m} \cdot \text{K} \cdot \text{s}} = \dfrac{\text{W}}{\text{m} \cdot \text{K}} = \text{W m}^{-1}\text{K}^{-1}$

since $\text{J/s} = \text{W}$ (watt).