'To look smb. in the eye' means 'to look squarely at someone', normally for the purpose of communicating something important, or perhaps to discern whether the other person is being truthful.
'To look into smb.'s eyes' implies that the action is carried out more in the manner of an inspection (perhaps in order to discern the presence or otherwise of romantic affection!), and typically lacks the implication of impending serious communication.