Is there a Biblical basis for the concept of being allowed to "be angry with God" or "tell him just how we feel"

There certainly is!

• 1 Peter 5:7 NLT Give all your worries and cares to God, for he cares about you.

I think anger falls under the category of "all worries and cares." Especially if you consider frustration a form of anger.

Take a look at these passages of David and Habakkuk expressing their frustration at God:

• Psalm 4:1-5 ESV Answer me when I call, O God of my righteousness! You have given me relief when I was in distress. Be gracious to me and hear my prayer! O men, how long shall my honor be turned into shame? How long will you love vain words and seek after lies? Selah But know that the Lord has set apart the godly for himself; the Lord hears when I call to him.

• Habakkuk 1:2-4 ESV O Lord, how long shall I cry for help, and you will not hear? Or cry to you “Violence!” and you will not save? Why do you make me see iniquity, and why do you idly look at wrong? Destruction and violence are before me; strife and contention arise. So the law is paralyzed, and justice never goes forth. For the wicked surround the righteous; so justice goes forth perverted.

• Psalm 13:1-2 ESV How long, O Lord? Will you forget me forever? How long will you hide your face from me? How long must I take counsel in my soul and have sorrow in my heart all the day? How long shall my enemy be exalted over me?

Job has been presented as a great example and an even more vivid example is David, a man God described as being after his own heart. A huge number of the Psalms of David consist of his weariness, frustration, and anger--these are known as the Imprecatory Psalms. There's debate as to whether we should pray for the destruction of our enemies as David did since Jesus says to love our enemies and pray for those that persecute us, but it's undeniable that the imprecatory psalms are soaked with anger. Read Psalm 4 above again and tell me that isn't a psalm of frustration.

Through all this it's important to remember not to act on anger towards other people.

• James 1:19,20 ESV Know this, my beloved brothers: let every person be quick to hear, slow to speak, slow to anger; for the anger of man does not produce the righteousness of God.
• Ephesians 4:25 ESV Be angry and do not sin; do not let the sun go down on your anger...
• Psalm 4:4-5 ESV Be angry, and do not sin; ponder in your own hearts on your beds, and be silent. Offer right sacrifices, and put your trust in the Lord.

But that's speaking for a human relations perspective. God can certainly handle our anger if we direct it towards him in desperation, and he wants us to. God wants our hearts--ALL of our heart!

Anger is probably the most primordial of man's imperfections.

Anger at God as proposed in you question would appear to fall under the following definition;

A violent passion of the mind excited by a real or supposed injury; usually accompanied with a  
propensity to take vengeance, or to obtain satisfaction from the offending party. This passion
however varies in degrees of violence, and in ingenuous minds, may be attended only with a desire
to reprove or chide the offender.

Anger at God is a slippery slope, most often when man becomes angry with God it is without justification. We tend to forget God does not serve us, but we are to serve God.

All Scripture is quoted from the King James translation.

Paul warned us about how easily we can slip from anger into sin in:

Ephesians 4:26 and 27 Be ye angry, and sin not: let not the sun go down upon your wrath: Neither give place to the devil.

We as mere mortals can never completely understand God nor his ways nor his plans. We must be ever wary of having our ire incite the anger of God which is always righteous.

It is always incumbent on us to remember that we are here only by the Grace of God, and should have by all rights been Destroyed both Body and Soul upon our first disobedience of God.

We can do worse than keeping these thing in mind, the next time we are tempted, by Satan to lash out at God.

Well, first consider the Commandment of Jesus himself: "You shall love the Lord your God with all your heart, and all your soul, and all your mind.". Unless we bring to God all of our emotions, including the unpleasant ones like despair, anger and doubt, we do not honestly and truly comply with the command to love with all of our heart.

As to examples of others who shared negative emotions with God,are the examples of Isaiah the Prophet who despaired

"Woe is me, for I am lost; for I am a man of unclean lips, and I dwell in the midst of a a people of unclean lips".;

Jonah the prophet, where ch 4 v 8b:

and [Jonah] asked that he might die, and said, "It is better for me to die than to live." But God said to Jonah, "Do you do well to be angry for the plant?" And [Jonah] said, "I do well to be angry, angry enough to die."; and

Elijah the Prophet, First Kings, 18:10ff:

"I have been very jealous for the Lord, the God of hosts; for the people of Israel have forsaken thy covenant, thrown down thy altars, and slain thy prophets with the sword, and I, even I only am am left, and they seek to take my life away."

And of course, as a true human, Jesus showed despair and anger with the Father in several places, and Paul urges us to be imitators of Christ.

Job.

Job, who is afflicted by The Accuser, spends most of the book shuttling between anger and faith. He remains steadfast, though he does not shy away from expressing his displeasure at the injustices heaped upon him.

His friends, who express all the typically pious nonsense you still hear today from Christians trying to be helpful, are of no help:

• You must have done something bad, or else this wouldn't have happened to you.
• God has a plan for your life.
• You need to repent.

All of their babbling misses the point, which is that suffering is inevitable, anger toward God is perfectly reasonable (He can take it), and that - FINALLY - the purposes of God are ultimately mysterious and behind our comprehension.

God reveals Himself to Job toward the end of the book, and delivers the longest divine monologue in the Scriptures - the most God has to say about His own nature anywhere in the Bible. And it's quite the speech: "Gird up your loins like a man, and face me." (Or, as I would have rendered it: Put on your BIG BOY panties, Job - things are getting real up in here.)

One way of looking at God's EPIC rebuke of Job might be that his reaction was blasphemous and he needed to be punished or corrected. But I find this a very immature and incomplete reading of the text, especially considering Job's ultimate vindication.

In my view, the better way of understanding this account is to realize that it is through the midst of Job's own dealing with his anger - honestly - with God, that he finally has a Theophanic (God Revealing) experience.

Job reveals his innermost self to God, and God responds by revealing Himself to Job.

You, too, can come before The Lord in your anger, or fear, or despair. You cannot hide from the almighty, whether you fly up to heaven or make your bed in hell. Only by offering your anger, yourself, to God will you ever be able to let go of it.

Your question has evolved to include "and let him know it."

Do not forget that God ALREADY KNOWS. Expressing your anger at God is not about informing the All-knowing Master of the Universe of something. It is, rather, about you coming to grips with your own anger in the presence of the Lord.