"look at someone unpleasantly, especially sexually" The noun leer originally meant "cheek" or "face", and from there developed the sense of "appearance". The modern sense of both noun and verb is always negative - it suggests a look that is sly or immodest, usually with a strong sexual element. We cannot leer innocently. But in Shakespeare's time, it had a neutral use too. The sexual sense is there in Merry Wives of Windsor (I.iii.41) when Falstaff says of Mistress Ford that "she gives the leer of invitation" (we would say "a come-hither look"). But the verb uses have no such suggestion. When Berowne says to Boyet (in Love's Labour's Lost, V.ii.480) "You leer upon me, do you?" he means no more than "cast a side glance". And when Falstaff tells Shallow that when King Henry V passes by "I will leer upon him" (Henry IV Part 2, V.v.6) it simply means "smile disarmingly".
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin