The covalent character of any compound can be predicted on the basis of Fajans' Rule.
Fajans' Rule, simply stated, says that a smaller cation can polarize a given anion better than a larger cation and that larger anions are more polarizable than smaller anions.
$\ce{Li+}$ being a small cation polarizes $\ce{Cl-}$ to a good extent and since any amount of polarization increases the covalent character of a compound, $\ce{LiCl}$ can be said to have some covalent nature.
$\ce{LiCl}$ being somewhat covalent dissolves in organic solvents(Remember, Like dissolves Like) such as ethanol, acetone, ethyl acetate and pyridine.
Further, since $\ce{Na+}$ is unable to polarize $\ce{Cl-}$ to a good degree it is regarded mainly as an ionic compound.