$PbCl _{2}$ react with $HCl$ as follows
$P b C l_{2}(s)+C l^{-} \xrightarrow{\text { Cold }}\left[P b C l_{3}\right]^{-}(a q) $
$P b C l_{2}(s)+2 C l^{-} \xrightarrow[\text{of HCl}]{\text { Excess }}\left[P b C l_{4}\right]^{2-}(a q)$
Thus, addition of excess of $C l^{-}$ions change the $P b C l_{2}$ as soluble complex of $\left[ PbCl _{4}\right]^{-2}$.
Hence, becomes soluble.