The Bible is really serious about this. How serious? Well, think of someone who really, really sinned badly against people — sexually used one, had another killed, betrayed the trust of scores and hundreds. Of course, you know I am thinking of King David. And you know right where I'm going, to his head-scratching confession in Psalm 51:4 — "Against you, you only, have I sinned and done what is evil in your sight." On the face of it, this is simply an absurd statement. "You only"? What of Bathsheba? What of his own wife/wives? What of Uriah?
Yet David knew the truth that most deeply offends atheists, humanists, everyone who's bought the big lie: sin is only sin because God defines it as sin. Sin is sin because God says it is sin. Spouse betraying spouse, children failing to honor parents, neighbor slandering or robbing neighbor — all these are sins because God says they are sins.
Apart from God, there would be no sin. All sin is, in the final analysis, against God; for without God there would be no sin. And so, as I have heard D. A. Carson say well more than once, in all sin, God is the primary offended party.
Back to the notion of forgiving yourself. It simply is faddish, man-exalting nonsense to speak of forgiving yourself. In your sin, you aren't the wronged party. If you (or I) really think that it is meaningful to speak of forgiving ourselves for our sins, then I don't think we've got that whole repentance thing straight. It isn't our own forgiveness which we need. It is, of course, God's forgiveness, a forgiveness that cost the Son of God His lifeblood (Ephesians 1:7; Hebrews 9:13-14, 22; 10:4).
It is only meaningful to speak of forgiveness of ourselves, then (A) by God, (B) of sin (C) against God, extended to us only (D) because of the shed blood of Christ, through whom alone we can find forgiveness. Secondarily, it is meaningful to speak of horizontal forgiveness of people upon repentance (that is not the focus of this post; but I can recommend a great book on the subject). But we know that God does command us to repent of our wrongs against others, to pursue restitution for those wrongs. We do those things because God calls us to do them. He defines my sin against my brother, He calls me to repent and pursue restitution and reconciliation (and commands me to forgive those who repent of sins against me). It's still all about God, and it's a subject of direct Biblical teaching.
Forgiving myself, however? Never. Biblically meaningless at best.