Charles' law is the change in volume with respect to temperature at constant pressure, while Boyle's law is the change in volume with respect to pressure at a constant temperature.
Taking a gas of volume $V_1$, pressure $P_1$ and temperature $T_1$, and let its change have a state $\left(V_2, P_2, T_2\right)$, then according to Boyle's law,
$P_1 V_1=P_2 V_2 \ldots \ldots . . . .(1)$
Then keeping this constant pressure, move to state $\left(V_2, P_2, T_2\right)$ using Charles' law,
$\frac{V_2}{T_1}=\frac{V_2}{T_2} \ldots \ldots . . . .(2)$
Solving for $V_2$, in both (1) and (2) and then equating them, we get-
$\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}$
After changing the pressure, volume and temperature of the gas, still their product is equal, suggesting that the relation is constant: $\frac{P V}{T}=k$.