When I think of a judge, I either think of a John Cleese-esk guy with a big gavel and funny wig straight out of the 1700's or think of Owen from Drop Dead Diva. However, this is not the image Peter is presenting to his audience. Instead of a stern and aloof judge, he presents a father. A good father bears the connotation of love, of gentleness and of protection not of condemnation and of judgment.
Peter's audience was facing religious persecution and they needed to understand who God is in light of their persecution. They were scattered throughout the Roman Empire, no longer living in the Promised Land but rather in ones consumed with cults and false gods everywhere you looked. It is into this situation that Peter says we have a Father who is our judge.
When you come before a judge, you state your accusations and leave the decision and punishment in his hands. He is the one who decides the outcome. We may not like the outcome, but it is decided; it is no longer in our hands. By presenting our case to our Father, it is easier to believe he will give us justice. Subsequently, this makes forgiveness easier as well.
We need to trust God will see justice through on our behalf according to His standard; when we do, we are satisfied with the judgment. We may not see it but God will bring justice. However, when we appeal a case brought to our Father, we are seeking a judgment from someone above God. Since the highest power is God, what results is bitterness and anger because we do not trust Him to bring justice on our behalf. We either do not accept his decision as just or do not believe He will act on our behalf.
As a side note: It can be hard not to push our perceptions of ourselves and others onto God. Let Him show you who He is. He longs to reveal Himself to you. Recently God has been doing a major upgrade in my understanding of who He is. He is taking away the perceptions I forced onto Him and is giving me a clearer understanding of who He actually is. He is taking away my perception of myself and showing me who I really am in Him. Sometimes we just need to ask our Father for He longs to reveal more of Himself to us (Acts 17:26-27).