Is it considered a lie to show that one has not forgiven family and relatives—who backbite and wrong the questioner—with the aim of deterring them, while the questioner has inwardly forgiven them?

Allah Almighty has promised forgiveness and a great reward to those who forgive people, as He says: "So whoever pardons and makes reconciliation - his reward is due from Allah," and His saying: "But whoever is patient and forgives, indeed, that is of the matters [requiring] determination," and His saying: "and those who restrain anger and who pardon the people." The Prophet, peace and blessings be upon him, gave glad tidings of abundant reward to whoever restrains his anger. As for your statement that you do not forgive people while you do forgive them, you can use tawriyah (double entendre) and ma'areedh (allusive speech) to avoid lying, by intending a meaning different from what you outwardly say.